Research ArticleArgument Properties for a Class of Analytic FunctionsInvolving Libera Transform

Badr S Alkahtani1 Teodor Bulboacs2 Rakesh Kumar3 and Rubayyi Alqahtani4

1Mathematics Department College of Science King Saud University Riyadh 11989 Saudi Arabia2Faculty of Mathematics and Computer Science Babes-Bolyai University 400084 Cluj-Napoca Romania3Department of Mathematics Amity University Rajasthan NH-11C Jaipur 302002 India4Department of Mathematics and Statistics College of Science Al-ImamMohammad Ibn Saud Islamic University (IMSIU)PO Box 65892 Riyadh 11566 Saudi Arabia

Correspondence should be addressed to Rakesh Kumar rkyadav11gmailcom

Received 10 June 2016 Accepted 4 September 2016

Academic Editor Jaeyoung Chung

Copyright copy 2016 Badr S Alkahtani et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The purpose of this paper is to find sufficient argument properties such that the images of some subclasses of functions by theLibera transform have bounded arguments

1 Introduction

LetA denote the class of functions of the form

119891 (119911) = 119911 + infinsum119899=2

119886119899119911119899 (1)

which are analytic in the open unit diskU = 119911 isin C |119911| lt 1A function 119891 isin A is said to be in the class Slowast(120572) of star-

like functions of order 120572 in U if and only if it satisfies thecondition

Re1199111198911015840 (119911)119891 (119911) gt 120572 119911 isin U (120572 lt 1) (2)

A function 119891 isin A is said to be in the class Klowast(120572) ofconvex functions of order 120572 inU if and only if it satisfies thecondition

Re(1 + 11991111989110158401015840 (119911)1198911015840 (119911) ) gt 120572 119911 isin U (120572 lt 1) (3)

It is well-known that Slowast(120572) sub SlowastK(120572) sub K whenever0 le 120572 lt 1 where Slowast equiv Slowast(0) and K equiv K(0) representrespectively the class of starlike and convex (normalized)functions

If 119891 isin A satisfies the condition

Re119891 (119911)1199111198911015840 (119911) gt 120573 119911 isin U (0 le 120573 lt 1) (4)

then 119891 is said to be star-like of reciprocal order 120573 and wedenote this class byNSlowast(120573)

The above definitionwas recently discussed byNunokawaet al [1] and Ravichandran and Sivaprasad Kumar [2]

Motivated by their works we define a class of convexfunctions of reciprocal order 120573 as follows

Definition 1 If 119891 isin A satisfies the condition

Re(1 + 11991111989110158401015840 (119911)1198911015840 (119911) )minus1 gt 120573 119911 isin U (0 le 120573 lt 1) (5)

then 119891 is said to be convex function of reciprocal order 120573 andwe denote this class byNK(120573)Definition 2 (see [3 4]) Suppose that119891 and119892 are two analyticfunctions inUWe say that the function119891 is subordinate to119892written 119891(119911) ≺ 119892(119911) if there exists a Schwarz function 119908 thatis a function 119908 analytic inU with 119908(0) = 0 and |119908(119911)| lt 1such that 119891(119911) = 119892(119908(119911)) 119911 isin U

Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 8908705 6 pageshttpdxdoiorg10115520168908705

2 Journal of Function Spaces

It is well-known that (see eg [3 4]) if 119892 is univalent inU then the following subordination property holds

119891 (119911) ≺ 119892 (119911) lArrrArr119891 (0) = 119892 (0) 119891 (U) sub 119892 (U) (6)

Remarks 1 (1) In view of the fact that

Re119901 (119911) gt 120573 119911 isin U (0 le 120573 lt 1)dArrRe 1119901 (119911) = Re

119901 (119911)1003816100381610038161003816119901 (119911)10038161003816100381610038162 gt 0 119911 isin U (7)

it follows that a star-like (convex) function of reciprocal order120573 is a star-like (convex) function Thus

NSlowast (120573) sub S

lowast (0) NK (120573) sub K (0)

(0 le 120573 lt 1) (8)

and the equality holds in both cases if and only if 120573 = 0(2) Let 0 lt 120573 lt 1 and suppose that 119891 isin A Then

(i)

119891 isin NSlowast (120573) lArrrArr100381610038161003816100381610038161003816100381610038161003816 119911119891

1015840 (119911)119891 (119911) minus 12120573 100381610038161003816100381610038161003816100381610038161003816 lt 12120573 119911 isin U (9)

(ii)

119891 isin NK (120573) lArrrArr1003816100381610038161003816100381610038161003816100381610038161 + 11991111989110158401015840 (119911)1198911015840 (119911) minus 12120573 100381610038161003816100381610038161003816100381610038161003816 lt 12120573 119911 isin U (10)

(3) We note that for 0 le 120573 lt 1 we have119891 isin NK (120573) lArrrArr

1199111198911015840 (119911) isin NSlowast (120573) (11)

In the following we give some examples of functionsbelonging to the class of star-like functions of reciprocal orderand the class of convex functions of reciprocal order

Example 3 (1) For 0 le 120573 lt 1 define the function119891120573 U rarr C

by

119891120573 (119911)=

119911 [1 + (1 minus 2120573) 119911]minus2(1minus120573)(1minus2120573) if 120573 isin [0 1) 12 119911119890minus119911 if 120573 = 12 (12)

Since 119891120573 isin A and a simple computation shows that

Re119891120573 (119911)1199111198911015840120573 (119911) = Re

1 + (1 minus 2120573) 1199111 minus 119911 gt 120573 119911 isin U (13)

we conclude that 119891120573 isin NSlowast(120573)(2) Using the third part of the Remarks 1 we deduce that

120593120573 isin NK (120573) lArrrArr119891120573 (119911) = 1199111205931015840120573 (119911) isin NS

lowast (120573) (14)

Since we proved that 119891120573 isin NSlowast(120573) and from the abovedifferential equation we obtain

120593120573 (119911)=

int1199110[1 + (1 minus 2120573) 119906]minus2(1minus120573)(1minus2120573) 119889119906 if 120573 isin [0 1) 12 1 minus 119890minus119911 if 120573 = 12

(15)

it follows that 120593120573 isin NK(120573)Example 4 (1) For 12 le 120573 lt 1 consider the function 120595120573 U rarr C defined by

120595120573 (119911) = 119911119890((1minus120573)120573)119911 (16)

This gives that

Re120595120573 (119911)1199111205951015840120573 (119911) = Re

120573120573 + (1 minus 120573) 119911 = Re ℎ (119911) 119911 isin U (17)

where

ℎ (119911) = 120573120573 + (1 minus 120573) 119911 (18)

Since Re120573 ge 12 it follows that ℎ is analytic inU Usingthe fact thatℎ is a convex (not necessary normalized) functioninU and ℎ(U) is symmetric with respect to the real axis wededuce that Re ℎ(119911) gt 120573 for all 119911 isin U hence 120595120573 isin NSlowast(120573)

(2) From the above result using the same reasons like inthe second part of Example 3 we obtain that

120594120573 (119911) = int1199110119890((1minus120573)120573)119906119889119906 isin NK (120573)

(12 le 120573 lt 1) (19)

We mention that several authors have investigated thestrongly star-like functions and the strongly convex functions(see [5ndash18]) In the present investigation we give some argu-ment properties of analytic functions belonging to A suchthat the images of these functions by the Libera transformhave bounded arguments

Journal of Function Spaces 3

2 Main Results

The following lemma will be used to prove our main results

Lemma 5 (see [16]) Let 1205720 be the solution of

120572120587 = 31205872 minus tanminus1120574 (20)

for a suitable fixed 120574 gt 0 so that 120582 U rarr C satisfies

120572Re 120582 (119911)1 + 120572 |Im 120582 (119911)| ge 120574 119911 isin U (21)

and let

120578 fl 120578 (120572 120574) = 120572 + 2120587 tanminus1120574 0 lt 120572 le 1205720 (22)

If 119901 is analytic inU 119901(0) = 1 and10038161003816100381610038161003816arg [119901 (119911) + 120582 (119911) 1199111199011015840 (119911)]10038161003816100381610038161003816 lt 1205872 120578 119911 isin U (23)

then 1003816100381610038161003816arg119901 (119911)1003816100381610038161003816 lt 1205872 120572 119911 isin U (24)

Remark that in the article [19] the author consideredsome special situations improving many results with a lot ofapplications Thus in [19 Lemma 2] the author proved thenext result

Let 120582 be a function defined onU satisfying1003816100381610038161003816arg (120582 (119911) minus 120578)1003816100381610038161003816 lt 1205873 119911 isin Usuch that 120578 = inf

119911isinU(Re 120582 (119911) minus |Im 120582 (119911)|radic3 ) (25)

and let

1205731015840 (120578) = (6 + 51205782 + 2radic9 + 151205782251205782 )13

sdot 9 minus 2radic9 + 15120578210(26)

be such that 21205731015840(120578) + 120578 ge 0 If 119901 is analytic in U 119901(0) = 1then

Re [119901 (119911) + 120582 (119911) 1199111199011015840 (119911)] gt 1205731015840 (120578) 119911 isin U

dArr1003816100381610038161003816arg119901 (119911)1003816100381610038161003816 lt 1205873 119911 isin U(27)

We emphasize that a special case of this lemma improvesthe conclusion of Lemma 1 from [19] and in the same articlethe author derived a number of interesting consequences ofit

LetL A rarr A defined by

L119891 (119911) = 2119911 int1199110 119891 (119905) 119889119905 (28)

be the well-known Libera transform [20]We first prove the following theorem which is essential

for proving our other results

Theorem 6 For 119892 isin A suppose that the Libera transform119866 =L119892 satisfies the condition (21) with

120582 (119911) = (2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 (29)

where 0 lt 120572 le 1205720 and 1205720 is defined by (20) If 119891 isin A and119865 = L119891 then100381610038161003816100381610038161003816100381610038161003816arg 1198911015840 (119911)1198921015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U

(30)

implies 100381610038161003816100381610038161003816100381610038161003816arg 1198651015840 (119911)1198661015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120572 119911 isin U (31)

Proof Let us define the function 119901 by

119901 (119911) = 1198651015840 (119911)1198661015840 (119911) 119911 isin U (32)

Suppose that there exists a number 1199110 isin U such that1198661015840(1199110) =0 since1198661015840(0) = 1 then 1199110 isin U0 It follows that there existsa unique number 119899 isin N and a unique function 119876 analytic inU with 119876(1199110) = 0 such that

1198661015840 (119911) = (119911 minus 1199110)119899 119876 (119911) 119911 isin U (33)

From the above relation we deduce that11991111986610158401015840 (119911)1198661015840 (119911) = 119899119911119911 minus 1199110 + 1199111198761015840 (119911)119876 (119911) 119911 isin U 1199110 (34)

which implies that 1199110 is a simple pole for the function11991111986610158401015840(119911)1198661015840(119911) Consequently from (29) we obtain that120582(1199110) = 0 and using the fact that 120574 satisfies the inequality(21) for all 119911 isin U it follows that

120574 le 120572Re 120582 (1199110)1 + 120572 1003816100381610038161003816Im 120582 (1199110)1003816100381610038161003816 = 0 (35)

which contradicts the assumption 120574 gt 0Thus 1198661015840(119911) = 0 for all 119911 isin U which implies that the

function 119901 is analytic inU and 119901(0) = 1Now a simple calculus shows that

1198911015840 (119911)1198921015840 (119911) = 11991111986510158401015840 (119911) + 21198651015840 (119911)11991111986610158401015840 (119911) + 21198661015840 (119911) = 119901 (119911) + 120582 (119911) 1199111199011015840 (119911) (36)

and our result follows immediately from Lemma 5

4 Journal of Function Spaces

Taking 119892(119911) = 119911 inTheorem 6 for the function 120582 definedby (29) we have 120582(119911) = 12 and we obtain the followingresult

Corollary 7 Suppose that the parameters 120572 and 120574 satisfy theconditions (20) and (22) for a given 120574 le 1205722 If 119891 isin A and119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (37)

implies 10038161003816100381610038161003816arg1198651015840 (z)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (38)

Example 8 For 119891 isin A and 119865 = L119891 considering the specialcases 120574 = 1radic3 120574 = radic2 minus 1 and 120574 = 2 minus radic3 from the abovecorollary we obtain respectively the next implications

(1) If 2radic3 le 120572 le 43 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 1radic3) = 1205872 (120572 + 13) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(39)

(2) If 2(radic2 minus 1) le 120572 le 118 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572radic2 minus 1) = 1205872 (120572 + 14) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(40)

(3) If 2(2 minus radic3) le 120572 le 1712 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(41)

Theorem 9 For 119892 isin A suppose that the Libera transform119866 =L119892 isin NK(120573) 0 lt 120573 lt 1 Suppose that 1205720 is defined by (20)and for 0 lt 120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le fl (120572 120573)

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 (42)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161003816100381610038161003816arg 1198911015840 (119911)1198921015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U

(43)

implies 100381610038161003816100381610038161003816100381610038161003816arg 1198651015840 (119911)1198661015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120572 119911 isin U (44)

Proof Let 119901 be the function defined by

119901 (119911) = 1198651015840 (119911)1198661015840 (119911) 119911 isin U (45)

Similarly like in the proof of Theorem 6 we will prove thatthe function 119901 is analytic inU Supposing that there exists anumber 1199110 isin U such that 1198661015840(1199110) = 0 since 1198661015840(0) = 1 then1199110 isin U 0 Hence there exists a unique number 119899 isin N anda unique function119876 analytic inU with119876(1199110) = 0 such that

1198661015840 (119911) = (119911 minus 1199110)119899 119876 (119911) 119911 isin U (46)

From here we deduce that

11991111986610158401015840 (119911)1198661015840 (119911) = 119899119911119911 minus 1199110 + 1199111198761015840 (119911)119876 (119911) 119911 isin U 1199110 (47)

which implies that 1199110 is a simple pole for the function11991111986610158401015840(119911)1198661015840(119911) Now using the fact that 119866 = L119892 isin NK(120573)we obtain

0 = Re(1 + 11991111986610158401015840 (1199110)1198661015840 (1199110) )minus1 gt 120573 (48)

which contradicts the assumption 0 le 120573Consequently 1198661015840(119911) = 0 for all 119911 isin U and this implies

that the function 119901 is analytic inU with 119901(0) = 1From (45) we easily get

1198911015840 (119911)1198921015840 (119911) = 119901 (119911) + 120582 (119911) 1199111199011015840 (119911) (49)

where the function 120582 is given by (29)The assumption 119866 = L119892 isin NK(120573) is equivalent to

(1 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 ≺ 1 + (1 minus 2120573) 1199111 minus 119911 (50)

hence there exists a Schwarz function 119908 such that

(1 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 = 1 + (1 minus 2120573)119908 (119911)1 minus 119908 (119911) 119911 isin U (51)

By simple calculations we get

(2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 = 1 + (1 minus 2120573)119908 (119911)2 (1 minus 120573119908 (119911)) 119911 isin U (52)

that is

120582 (119911) = (2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 ≺ 1 + (1 minus 2120573) 1199112 (1 minus 120573119911) š 120601 (119911) (53)

Journal of Function Spaces 5

Since the circular transform 120601maps the unit diskU onto thediskΩ = 119908 isin C |119908 minus 120596| lt 119903

with 120596 = 1 + 21205732 (1 + 120573) 119903 = 12 (1 + 120573) (54)

from the subordination property (6) we deduce that thesubordination relation (53) is equivalent to

120582 (119911) = 119906 (119911) + 119894V (119911) isin Ω 119911 isin U (55)

With the above notation the condition (21) becomes

120572119906 (119911) minus 120572120574 |V (119911)| minus 120574 ge 0 119911 isin U (56)

We will prove that if 120574 satisfies the assumption (42) thatis 0 lt 120574 le then the subordination (53) implies that theinequality (56) holds that is

Ω sub Δ (57)

where

Δ = 119908 = 119906 + 119894V isin C 120572119906 minus 120572120574 |V| minus 120574 ge 0 (58)

Since the setsΩ and Δ are symmetric with respect to the realaxis the above inclusion is equivalent to

Ω sub Δ (59)

where

Ω = 119908 = 119906 + 119894V isin Ω V ge 0 Δ = 119908 = 119906 + 119894V isin Δ V ge 0 (60)

The line (119889) containing the half-line120572119906 minus 120572120574V minus 120574 = 0 V ge 0 (61)

which is a part of the boundary the set Δ has the equation120572119906 minus 120572120574V minus 120574 = 0 (62)

Thus the line (119889) intersects the real axis in the point 120574120572and contains the point minus119894120572 (see Figure 1) The smallest setΔ that includes the domain Ω is obtained for the case whenthe line (119889) containing the point minus119894120572 is tangent to the upperboundary of Ω which is the half-circle

(C) = 119908 = 119906 + 119894V isin C (119906 minus 120596)2 + V2 = 1199032 V ge 0 (63)

In this case as it is shown in the figure the line (119889) becomesthe tangent line (119889120591) to (C)

We will determine now the equation of the tangent line(119889120591) If we consider the family (119889119898) of all the lines containingthe point minus119894120572 with nonnegative angular coefficient that is

(119889119898) V = 119898119906 minus 1120572 119898 isin (0 +infin) (64)

u1

v

O

(d) (d120591)

Δ

Ω

120574

120572

120574

120572minus1

120572i

120573

1 + 120573

Figure 1 The inclusion Ω sub Δwe will solve the system

(119906 minus 120596)2 + V2 = 1199032 V ge 0V = 119898119906 minus 1120572 (65)

It follows that

1205722 (1 + 1198982) 1199062 minus 2120572 (120572120596 + 119898) 119906 + 1205722 (1205962 minus 1199032) + 1= 0 (66)

and the line is tangent to the half-circle if and only if thediscriminant of the above quadratic form is zero that is

1205722 (1205962 minus 1199032)1198982 minus 2120572120596119898 minus 12057221199032 + 1 = 0 (67)

This last equation has the biggest positive root

= 120596 + 119903radic(1205962 minus 1199032) 1205722 + 1120572 (1205962 minus 1199032) (68)

hence the tangent line to the half-circle (C) which containsthe point minus119894120572 has the equation

(119889120591) V = 119906 minus 1120572 (69)

The tangent (119889120591) intersects the real axis in the point 1(120572)hence we deduce that the inclusion (59) holds if and only if

0 lt 120574120572 le 1120572 (70)

that is

0 lt 120574 le 1119898 = 120572 (1205962 minus 1199032)120596 + 119903radic(1205962 minus 1199032) 1205722 + 1

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 = (120572 120573) (71)

Thus if the parameter 120574 satisfies the assumption (42)then the condition (21) of Lemma 5 is fulfilled and our resultfollows fromTheorem 6

6 Journal of Function Spaces

Putting 119892(119911) = 119911 in Theorem 9 for 120573 rarr 1minus we get thenext result

Corollary 10 Suppose that 1205720 is defined by (20) and for 0 lt120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le 21205723 + radic12057222 + 1 (72)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (73)

implies 10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (74)

Example 11 For 119891 isin A and 119865 = L119891 considering the specialcase 120574 = 2 minus radic3 from Corollary 10 we deduce the nextimplication

If

054552 ≃ minus168 + 108radic3 + 164radic2 minus 92radic647 le 120572le 1712

(75)

then 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(76)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Dr Badr S Alkahtani andDr Rubayyi Alqahtani extend theirsincere appropriations to the Deanship of Scientific Researchat King Saud University for funding this Profile ResearchGroup (PRG-1437-35)

References

[1] M Nunokawa S Owa J Nishiwaki K Kuroki and T HayamildquoDifferential subordination and argumental propertyrdquoComput-ers amp Mathematics with Applications vol 56 no 10 pp 2733ndash2736 2008

[2] V Ravichandran and S Sivaprasad Kumar ldquoArgument estimatefor starlike functions of reciprocal orderrdquo Southeast AsianBulletin of Mathematics vol 35 no 5 pp 837ndash843 2011

[3] C Pommerenke Univalent Functions Vanderhoeck and Ru-precht Gottingen Germany 1975

[4] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[5] M K Aouf J Dziok and J Sokol ldquoOn a subclass of stronglystarlike functionsrdquo Applied Mathematics Letters vol 24 no 1pp 27ndash32 2011

[6] N E Cho ldquoArgument estimates of certain meromorphic func-tionsrdquo Communications Korean Mathematical Society vol 15no 2 pp 263ndash274 2000

[7] N E Cho and J A Kim ldquoAngular estimations of certain analyticfunctionsrdquo Journal of the Korean Mathematical Society vol 34no 2 pp 427ndash436 1997

[8] N E Cho Y C Kim andHM Srivastava ldquoArgument estimatesfor a certain class of analytic functionsrdquo Complex VariablesTheory and Application vol 38 no 3 pp 277ndash287 1999

[9] N E Cho J Patel and G P Mohapatra ldquoArgument estimates ofcertain multivalent functions involving a linear operatorrdquo Inter-national Journal of Mathematics andMathematical Sciences vol31 no 11 pp 659ndash673 2002

[10] N E Cho and H M Srivastava ldquoArgument estimates of certainanalytic functions defined by a class of multiplier transforma-tionsrdquo Mathematical and Computer Modelling vol 37 no 1-2pp 39ndash49 2003

[11] N ECho S YWoo and SOwa ldquoArgument estimates of certainmeromorphic functionsrdquo Mathematica Balkanica New Seriesvol 15 no 3-4 pp 285ndash295 2001

[12] S Kanas andT Sugawa ldquoStrong starlikeness for a class of convexfunctionsrdquo Journal of Mathematical Analysis and Applicationsvol 336 no 2 pp 1005ndash1017 2007

[13] MNunokawa ldquoOn the order of strongly starlikeness of stronglyconvex functionsrdquo Japan Academy Proceedings Series A Math-ematical Sciences vol 69 no 7 pp 234ndash237 1993

[14] M Nunokawa S Owa H Saitoh and N N Pascu ldquoArgumentestimates for certain analytic functionsrdquo Proceedings of theJapan Academy Series A Mathematical Sciences vol 79 no 10pp 163ndash166 2003

[15] C Pommerenke ldquoOn close-to-convex analytic functionsrdquoTransactions of the American Mathematical Society vol 114 pp176ndash186 1965

[16] S Ponnusamy ldquoDifferential subordinations concerning starlikefunctionsrdquo Proceedings of the Indian Academy of SciencesmdashMathematical Sciences vol 104 no 2 pp 397ndash411 1994

[17] J K Prajapat and S P Goyal ldquoApplications of Srivastava-Attiyaoperator to the classes of strongly starlike and strongly convexfunctionsrdquo Journal of Mathematical Inequalities vol 3 no 1 pp129ndash137 2009

[18] V RavichandranM Darus andN Seenivasagan ldquoOn a criteriafor strong starlikenessrdquo Australian Journal of MathematicalAnalysis and Applications vol 2 no 1 article 6 12 pages 2005

[19] S Ponnusamy ldquoDifferential subordination and Bazilevic func-tionsrdquoProceedings of the IndianAcademy of SciencesMathemat-ical Sciences vol 105 no 2 pp 169ndash186 1995

[20] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

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Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

It is well-known that (see eg [3 4]) if 119892 is univalent inU then the following subordination property holds

119891 (119911) ≺ 119892 (119911) lArrrArr119891 (0) = 119892 (0) 119891 (U) sub 119892 (U) (6)

Remarks 1 (1) In view of the fact that

Re119901 (119911) gt 120573 119911 isin U (0 le 120573 lt 1)dArrRe 1119901 (119911) = Re

119901 (119911)1003816100381610038161003816119901 (119911)10038161003816100381610038162 gt 0 119911 isin U (7)

it follows that a star-like (convex) function of reciprocal order120573 is a star-like (convex) function Thus

NSlowast (120573) sub S

lowast (0) NK (120573) sub K (0)

(0 le 120573 lt 1) (8)

and the equality holds in both cases if and only if 120573 = 0(2) Let 0 lt 120573 lt 1 and suppose that 119891 isin A Then

(i)

119891 isin NSlowast (120573) lArrrArr100381610038161003816100381610038161003816100381610038161003816 119911119891

1015840 (119911)119891 (119911) minus 12120573 100381610038161003816100381610038161003816100381610038161003816 lt 12120573 119911 isin U (9)

(ii)

119891 isin NK (120573) lArrrArr1003816100381610038161003816100381610038161003816100381610038161 + 11991111989110158401015840 (119911)1198911015840 (119911) minus 12120573 100381610038161003816100381610038161003816100381610038161003816 lt 12120573 119911 isin U (10)

(3) We note that for 0 le 120573 lt 1 we have119891 isin NK (120573) lArrrArr

1199111198911015840 (119911) isin NSlowast (120573) (11)

In the following we give some examples of functionsbelonging to the class of star-like functions of reciprocal orderand the class of convex functions of reciprocal order

Example 3 (1) For 0 le 120573 lt 1 define the function119891120573 U rarr C

by

119891120573 (119911)=

119911 [1 + (1 minus 2120573) 119911]minus2(1minus120573)(1minus2120573) if 120573 isin [0 1) 12 119911119890minus119911 if 120573 = 12 (12)

Since 119891120573 isin A and a simple computation shows that

Re119891120573 (119911)1199111198911015840120573 (119911) = Re

1 + (1 minus 2120573) 1199111 minus 119911 gt 120573 119911 isin U (13)

we conclude that 119891120573 isin NSlowast(120573)(2) Using the third part of the Remarks 1 we deduce that

120593120573 isin NK (120573) lArrrArr119891120573 (119911) = 1199111205931015840120573 (119911) isin NS

lowast (120573) (14)

Since we proved that 119891120573 isin NSlowast(120573) and from the abovedifferential equation we obtain

120593120573 (119911)=

int1199110[1 + (1 minus 2120573) 119906]minus2(1minus120573)(1minus2120573) 119889119906 if 120573 isin [0 1) 12 1 minus 119890minus119911 if 120573 = 12

(15)

it follows that 120593120573 isin NK(120573)Example 4 (1) For 12 le 120573 lt 1 consider the function 120595120573 U rarr C defined by

120595120573 (119911) = 119911119890((1minus120573)120573)119911 (16)

This gives that

Re120595120573 (119911)1199111205951015840120573 (119911) = Re

120573120573 + (1 minus 120573) 119911 = Re ℎ (119911) 119911 isin U (17)

where

ℎ (119911) = 120573120573 + (1 minus 120573) 119911 (18)

Since Re120573 ge 12 it follows that ℎ is analytic inU Usingthe fact thatℎ is a convex (not necessary normalized) functioninU and ℎ(U) is symmetric with respect to the real axis wededuce that Re ℎ(119911) gt 120573 for all 119911 isin U hence 120595120573 isin NSlowast(120573)

(2) From the above result using the same reasons like inthe second part of Example 3 we obtain that

120594120573 (119911) = int1199110119890((1minus120573)120573)119906119889119906 isin NK (120573)

(12 le 120573 lt 1) (19)

We mention that several authors have investigated thestrongly star-like functions and the strongly convex functions(see [5ndash18]) In the present investigation we give some argu-ment properties of analytic functions belonging to A suchthat the images of these functions by the Libera transformhave bounded arguments

Journal of Function Spaces 3

2 Main Results

The following lemma will be used to prove our main results

Lemma 5 (see [16]) Let 1205720 be the solution of

120572120587 = 31205872 minus tanminus1120574 (20)

for a suitable fixed 120574 gt 0 so that 120582 U rarr C satisfies

120572Re 120582 (119911)1 + 120572 |Im 120582 (119911)| ge 120574 119911 isin U (21)

and let

120578 fl 120578 (120572 120574) = 120572 + 2120587 tanminus1120574 0 lt 120572 le 1205720 (22)

If 119901 is analytic inU 119901(0) = 1 and10038161003816100381610038161003816arg [119901 (119911) + 120582 (119911) 1199111199011015840 (119911)]10038161003816100381610038161003816 lt 1205872 120578 119911 isin U (23)

then 1003816100381610038161003816arg119901 (119911)1003816100381610038161003816 lt 1205872 120572 119911 isin U (24)

Remark that in the article [19] the author consideredsome special situations improving many results with a lot ofapplications Thus in [19 Lemma 2] the author proved thenext result

Let 120582 be a function defined onU satisfying1003816100381610038161003816arg (120582 (119911) minus 120578)1003816100381610038161003816 lt 1205873 119911 isin Usuch that 120578 = inf

119911isinU(Re 120582 (119911) minus |Im 120582 (119911)|radic3 ) (25)

and let

1205731015840 (120578) = (6 + 51205782 + 2radic9 + 151205782251205782 )13

sdot 9 minus 2radic9 + 15120578210(26)

be such that 21205731015840(120578) + 120578 ge 0 If 119901 is analytic in U 119901(0) = 1then

Re [119901 (119911) + 120582 (119911) 1199111199011015840 (119911)] gt 1205731015840 (120578) 119911 isin U

dArr1003816100381610038161003816arg119901 (119911)1003816100381610038161003816 lt 1205873 119911 isin U(27)

We emphasize that a special case of this lemma improvesthe conclusion of Lemma 1 from [19] and in the same articlethe author derived a number of interesting consequences ofit

LetL A rarr A defined by

L119891 (119911) = 2119911 int1199110 119891 (119905) 119889119905 (28)

be the well-known Libera transform [20]We first prove the following theorem which is essential

for proving our other results

Theorem 6 For 119892 isin A suppose that the Libera transform119866 =L119892 satisfies the condition (21) with

120582 (119911) = (2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 (29)

where 0 lt 120572 le 1205720 and 1205720 is defined by (20) If 119891 isin A and119865 = L119891 then100381610038161003816100381610038161003816100381610038161003816arg 1198911015840 (119911)1198921015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U

(30)

implies 100381610038161003816100381610038161003816100381610038161003816arg 1198651015840 (119911)1198661015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120572 119911 isin U (31)

Proof Let us define the function 119901 by

119901 (119911) = 1198651015840 (119911)1198661015840 (119911) 119911 isin U (32)

Suppose that there exists a number 1199110 isin U such that1198661015840(1199110) =0 since1198661015840(0) = 1 then 1199110 isin U0 It follows that there existsa unique number 119899 isin N and a unique function 119876 analytic inU with 119876(1199110) = 0 such that

1198661015840 (119911) = (119911 minus 1199110)119899 119876 (119911) 119911 isin U (33)

From the above relation we deduce that11991111986610158401015840 (119911)1198661015840 (119911) = 119899119911119911 minus 1199110 + 1199111198761015840 (119911)119876 (119911) 119911 isin U 1199110 (34)

which implies that 1199110 is a simple pole for the function11991111986610158401015840(119911)1198661015840(119911) Consequently from (29) we obtain that120582(1199110) = 0 and using the fact that 120574 satisfies the inequality(21) for all 119911 isin U it follows that

120574 le 120572Re 120582 (1199110)1 + 120572 1003816100381610038161003816Im 120582 (1199110)1003816100381610038161003816 = 0 (35)

which contradicts the assumption 120574 gt 0Thus 1198661015840(119911) = 0 for all 119911 isin U which implies that the

function 119901 is analytic inU and 119901(0) = 1Now a simple calculus shows that

1198911015840 (119911)1198921015840 (119911) = 11991111986510158401015840 (119911) + 21198651015840 (119911)11991111986610158401015840 (119911) + 21198661015840 (119911) = 119901 (119911) + 120582 (119911) 1199111199011015840 (119911) (36)

and our result follows immediately from Lemma 5

4 Journal of Function Spaces

Taking 119892(119911) = 119911 inTheorem 6 for the function 120582 definedby (29) we have 120582(119911) = 12 and we obtain the followingresult

Corollary 7 Suppose that the parameters 120572 and 120574 satisfy theconditions (20) and (22) for a given 120574 le 1205722 If 119891 isin A and119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (37)

implies 10038161003816100381610038161003816arg1198651015840 (z)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (38)

Example 8 For 119891 isin A and 119865 = L119891 considering the specialcases 120574 = 1radic3 120574 = radic2 minus 1 and 120574 = 2 minus radic3 from the abovecorollary we obtain respectively the next implications

(1) If 2radic3 le 120572 le 43 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 1radic3) = 1205872 (120572 + 13) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(39)

(2) If 2(radic2 minus 1) le 120572 le 118 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572radic2 minus 1) = 1205872 (120572 + 14) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(40)

(3) If 2(2 minus radic3) le 120572 le 1712 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(41)

Theorem 9 For 119892 isin A suppose that the Libera transform119866 =L119892 isin NK(120573) 0 lt 120573 lt 1 Suppose that 1205720 is defined by (20)and for 0 lt 120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le fl (120572 120573)

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 (42)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161003816100381610038161003816arg 1198911015840 (119911)1198921015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U

(43)

implies 100381610038161003816100381610038161003816100381610038161003816arg 1198651015840 (119911)1198661015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120572 119911 isin U (44)

Proof Let 119901 be the function defined by

119901 (119911) = 1198651015840 (119911)1198661015840 (119911) 119911 isin U (45)

Similarly like in the proof of Theorem 6 we will prove thatthe function 119901 is analytic inU Supposing that there exists anumber 1199110 isin U such that 1198661015840(1199110) = 0 since 1198661015840(0) = 1 then1199110 isin U 0 Hence there exists a unique number 119899 isin N anda unique function119876 analytic inU with119876(1199110) = 0 such that

1198661015840 (119911) = (119911 minus 1199110)119899 119876 (119911) 119911 isin U (46)

From here we deduce that

11991111986610158401015840 (119911)1198661015840 (119911) = 119899119911119911 minus 1199110 + 1199111198761015840 (119911)119876 (119911) 119911 isin U 1199110 (47)

which implies that 1199110 is a simple pole for the function11991111986610158401015840(119911)1198661015840(119911) Now using the fact that 119866 = L119892 isin NK(120573)we obtain

0 = Re(1 + 11991111986610158401015840 (1199110)1198661015840 (1199110) )minus1 gt 120573 (48)

which contradicts the assumption 0 le 120573Consequently 1198661015840(119911) = 0 for all 119911 isin U and this implies

that the function 119901 is analytic inU with 119901(0) = 1From (45) we easily get

1198911015840 (119911)1198921015840 (119911) = 119901 (119911) + 120582 (119911) 1199111199011015840 (119911) (49)

where the function 120582 is given by (29)The assumption 119866 = L119892 isin NK(120573) is equivalent to

(1 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 ≺ 1 + (1 minus 2120573) 1199111 minus 119911 (50)

hence there exists a Schwarz function 119908 such that

(1 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 = 1 + (1 minus 2120573)119908 (119911)1 minus 119908 (119911) 119911 isin U (51)

By simple calculations we get

(2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 = 1 + (1 minus 2120573)119908 (119911)2 (1 minus 120573119908 (119911)) 119911 isin U (52)

that is

120582 (119911) = (2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 ≺ 1 + (1 minus 2120573) 1199112 (1 minus 120573119911) š 120601 (119911) (53)

Journal of Function Spaces 5

Since the circular transform 120601maps the unit diskU onto thediskΩ = 119908 isin C |119908 minus 120596| lt 119903

with 120596 = 1 + 21205732 (1 + 120573) 119903 = 12 (1 + 120573) (54)

from the subordination property (6) we deduce that thesubordination relation (53) is equivalent to

120582 (119911) = 119906 (119911) + 119894V (119911) isin Ω 119911 isin U (55)

With the above notation the condition (21) becomes

120572119906 (119911) minus 120572120574 |V (119911)| minus 120574 ge 0 119911 isin U (56)

We will prove that if 120574 satisfies the assumption (42) thatis 0 lt 120574 le then the subordination (53) implies that theinequality (56) holds that is

Ω sub Δ (57)

where

Δ = 119908 = 119906 + 119894V isin C 120572119906 minus 120572120574 |V| minus 120574 ge 0 (58)

Since the setsΩ and Δ are symmetric with respect to the realaxis the above inclusion is equivalent to

Ω sub Δ (59)

where

Ω = 119908 = 119906 + 119894V isin Ω V ge 0 Δ = 119908 = 119906 + 119894V isin Δ V ge 0 (60)

The line (119889) containing the half-line120572119906 minus 120572120574V minus 120574 = 0 V ge 0 (61)

which is a part of the boundary the set Δ has the equation120572119906 minus 120572120574V minus 120574 = 0 (62)

Thus the line (119889) intersects the real axis in the point 120574120572and contains the point minus119894120572 (see Figure 1) The smallest setΔ that includes the domain Ω is obtained for the case whenthe line (119889) containing the point minus119894120572 is tangent to the upperboundary of Ω which is the half-circle

(C) = 119908 = 119906 + 119894V isin C (119906 minus 120596)2 + V2 = 1199032 V ge 0 (63)

In this case as it is shown in the figure the line (119889) becomesthe tangent line (119889120591) to (C)

We will determine now the equation of the tangent line(119889120591) If we consider the family (119889119898) of all the lines containingthe point minus119894120572 with nonnegative angular coefficient that is

(119889119898) V = 119898119906 minus 1120572 119898 isin (0 +infin) (64)

u1

v

O

(d) (d120591)

Δ

Ω

120574

120572

120574

120572minus1

120572i

120573

1 + 120573

Figure 1 The inclusion Ω sub Δwe will solve the system

(119906 minus 120596)2 + V2 = 1199032 V ge 0V = 119898119906 minus 1120572 (65)

It follows that

1205722 (1 + 1198982) 1199062 minus 2120572 (120572120596 + 119898) 119906 + 1205722 (1205962 minus 1199032) + 1= 0 (66)

and the line is tangent to the half-circle if and only if thediscriminant of the above quadratic form is zero that is

1205722 (1205962 minus 1199032)1198982 minus 2120572120596119898 minus 12057221199032 + 1 = 0 (67)

This last equation has the biggest positive root

= 120596 + 119903radic(1205962 minus 1199032) 1205722 + 1120572 (1205962 minus 1199032) (68)

hence the tangent line to the half-circle (C) which containsthe point minus119894120572 has the equation

(119889120591) V = 119906 minus 1120572 (69)

The tangent (119889120591) intersects the real axis in the point 1(120572)hence we deduce that the inclusion (59) holds if and only if

0 lt 120574120572 le 1120572 (70)

that is

0 lt 120574 le 1119898 = 120572 (1205962 minus 1199032)120596 + 119903radic(1205962 minus 1199032) 1205722 + 1

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 = (120572 120573) (71)

Thus if the parameter 120574 satisfies the assumption (42)then the condition (21) of Lemma 5 is fulfilled and our resultfollows fromTheorem 6

6 Journal of Function Spaces

Putting 119892(119911) = 119911 in Theorem 9 for 120573 rarr 1minus we get thenext result

Corollary 10 Suppose that 1205720 is defined by (20) and for 0 lt120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le 21205723 + radic12057222 + 1 (72)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (73)

implies 10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (74)

Example 11 For 119891 isin A and 119865 = L119891 considering the specialcase 120574 = 2 minus radic3 from Corollary 10 we deduce the nextimplication

If

054552 ≃ minus168 + 108radic3 + 164radic2 minus 92radic647 le 120572le 1712

(75)

then 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(76)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Dr Badr S Alkahtani andDr Rubayyi Alqahtani extend theirsincere appropriations to the Deanship of Scientific Researchat King Saud University for funding this Profile ResearchGroup (PRG-1437-35)

References

[1] M Nunokawa S Owa J Nishiwaki K Kuroki and T HayamildquoDifferential subordination and argumental propertyrdquoComput-ers amp Mathematics with Applications vol 56 no 10 pp 2733ndash2736 2008

[2] V Ravichandran and S Sivaprasad Kumar ldquoArgument estimatefor starlike functions of reciprocal orderrdquo Southeast AsianBulletin of Mathematics vol 35 no 5 pp 837ndash843 2011

[3] C Pommerenke Univalent Functions Vanderhoeck and Ru-precht Gottingen Germany 1975

[4] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[5] M K Aouf J Dziok and J Sokol ldquoOn a subclass of stronglystarlike functionsrdquo Applied Mathematics Letters vol 24 no 1pp 27ndash32 2011

[6] N E Cho ldquoArgument estimates of certain meromorphic func-tionsrdquo Communications Korean Mathematical Society vol 15no 2 pp 263ndash274 2000

[7] N E Cho and J A Kim ldquoAngular estimations of certain analyticfunctionsrdquo Journal of the Korean Mathematical Society vol 34no 2 pp 427ndash436 1997

[8] N E Cho Y C Kim andHM Srivastava ldquoArgument estimatesfor a certain class of analytic functionsrdquo Complex VariablesTheory and Application vol 38 no 3 pp 277ndash287 1999

[9] N E Cho J Patel and G P Mohapatra ldquoArgument estimates ofcertain multivalent functions involving a linear operatorrdquo Inter-national Journal of Mathematics andMathematical Sciences vol31 no 11 pp 659ndash673 2002

[10] N E Cho and H M Srivastava ldquoArgument estimates of certainanalytic functions defined by a class of multiplier transforma-tionsrdquo Mathematical and Computer Modelling vol 37 no 1-2pp 39ndash49 2003

[11] N ECho S YWoo and SOwa ldquoArgument estimates of certainmeromorphic functionsrdquo Mathematica Balkanica New Seriesvol 15 no 3-4 pp 285ndash295 2001

[12] S Kanas andT Sugawa ldquoStrong starlikeness for a class of convexfunctionsrdquo Journal of Mathematical Analysis and Applicationsvol 336 no 2 pp 1005ndash1017 2007

[13] MNunokawa ldquoOn the order of strongly starlikeness of stronglyconvex functionsrdquo Japan Academy Proceedings Series A Math-ematical Sciences vol 69 no 7 pp 234ndash237 1993

[14] M Nunokawa S Owa H Saitoh and N N Pascu ldquoArgumentestimates for certain analytic functionsrdquo Proceedings of theJapan Academy Series A Mathematical Sciences vol 79 no 10pp 163ndash166 2003

[15] C Pommerenke ldquoOn close-to-convex analytic functionsrdquoTransactions of the American Mathematical Society vol 114 pp176ndash186 1965

[16] S Ponnusamy ldquoDifferential subordinations concerning starlikefunctionsrdquo Proceedings of the Indian Academy of SciencesmdashMathematical Sciences vol 104 no 2 pp 397ndash411 1994

[17] J K Prajapat and S P Goyal ldquoApplications of Srivastava-Attiyaoperator to the classes of strongly starlike and strongly convexfunctionsrdquo Journal of Mathematical Inequalities vol 3 no 1 pp129ndash137 2009

[18] V RavichandranM Darus andN Seenivasagan ldquoOn a criteriafor strong starlikenessrdquo Australian Journal of MathematicalAnalysis and Applications vol 2 no 1 article 6 12 pages 2005

[19] S Ponnusamy ldquoDifferential subordination and Bazilevic func-tionsrdquoProceedings of the IndianAcademy of SciencesMathemat-ical Sciences vol 105 no 2 pp 169ndash186 1995

[20] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 3

2 Main Results

The following lemma will be used to prove our main results

Lemma 5 (see [16]) Let 1205720 be the solution of

120572120587 = 31205872 minus tanminus1120574 (20)

for a suitable fixed 120574 gt 0 so that 120582 U rarr C satisfies

120572Re 120582 (119911)1 + 120572 |Im 120582 (119911)| ge 120574 119911 isin U (21)

and let

120578 fl 120578 (120572 120574) = 120572 + 2120587 tanminus1120574 0 lt 120572 le 1205720 (22)

If 119901 is analytic inU 119901(0) = 1 and10038161003816100381610038161003816arg [119901 (119911) + 120582 (119911) 1199111199011015840 (119911)]10038161003816100381610038161003816 lt 1205872 120578 119911 isin U (23)

then 1003816100381610038161003816arg119901 (119911)1003816100381610038161003816 lt 1205872 120572 119911 isin U (24)

Remark that in the article [19] the author consideredsome special situations improving many results with a lot ofapplications Thus in [19 Lemma 2] the author proved thenext result

Let 120582 be a function defined onU satisfying1003816100381610038161003816arg (120582 (119911) minus 120578)1003816100381610038161003816 lt 1205873 119911 isin Usuch that 120578 = inf

119911isinU(Re 120582 (119911) minus |Im 120582 (119911)|radic3 ) (25)

and let

1205731015840 (120578) = (6 + 51205782 + 2radic9 + 151205782251205782 )13

sdot 9 minus 2radic9 + 15120578210(26)

be such that 21205731015840(120578) + 120578 ge 0 If 119901 is analytic in U 119901(0) = 1then

Re [119901 (119911) + 120582 (119911) 1199111199011015840 (119911)] gt 1205731015840 (120578) 119911 isin U

dArr1003816100381610038161003816arg119901 (119911)1003816100381610038161003816 lt 1205873 119911 isin U(27)

We emphasize that a special case of this lemma improvesthe conclusion of Lemma 1 from [19] and in the same articlethe author derived a number of interesting consequences ofit

LetL A rarr A defined by

L119891 (119911) = 2119911 int1199110 119891 (119905) 119889119905 (28)

be the well-known Libera transform [20]We first prove the following theorem which is essential

for proving our other results

Theorem 6 For 119892 isin A suppose that the Libera transform119866 =L119892 satisfies the condition (21) with

120582 (119911) = (2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 (29)

where 0 lt 120572 le 1205720 and 1205720 is defined by (20) If 119891 isin A and119865 = L119891 then100381610038161003816100381610038161003816100381610038161003816arg 1198911015840 (119911)1198921015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U

(30)

implies 100381610038161003816100381610038161003816100381610038161003816arg 1198651015840 (119911)1198661015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120572 119911 isin U (31)

Proof Let us define the function 119901 by

119901 (119911) = 1198651015840 (119911)1198661015840 (119911) 119911 isin U (32)

Suppose that there exists a number 1199110 isin U such that1198661015840(1199110) =0 since1198661015840(0) = 1 then 1199110 isin U0 It follows that there existsa unique number 119899 isin N and a unique function 119876 analytic inU with 119876(1199110) = 0 such that

1198661015840 (119911) = (119911 minus 1199110)119899 119876 (119911) 119911 isin U (33)

From the above relation we deduce that11991111986610158401015840 (119911)1198661015840 (119911) = 119899119911119911 minus 1199110 + 1199111198761015840 (119911)119876 (119911) 119911 isin U 1199110 (34)

which implies that 1199110 is a simple pole for the function11991111986610158401015840(119911)1198661015840(119911) Consequently from (29) we obtain that120582(1199110) = 0 and using the fact that 120574 satisfies the inequality(21) for all 119911 isin U it follows that

120574 le 120572Re 120582 (1199110)1 + 120572 1003816100381610038161003816Im 120582 (1199110)1003816100381610038161003816 = 0 (35)

which contradicts the assumption 120574 gt 0Thus 1198661015840(119911) = 0 for all 119911 isin U which implies that the

function 119901 is analytic inU and 119901(0) = 1Now a simple calculus shows that

1198911015840 (119911)1198921015840 (119911) = 11991111986510158401015840 (119911) + 21198651015840 (119911)11991111986610158401015840 (119911) + 21198661015840 (119911) = 119901 (119911) + 120582 (119911) 1199111199011015840 (119911) (36)

and our result follows immediately from Lemma 5

4 Journal of Function Spaces

Taking 119892(119911) = 119911 inTheorem 6 for the function 120582 definedby (29) we have 120582(119911) = 12 and we obtain the followingresult

Corollary 7 Suppose that the parameters 120572 and 120574 satisfy theconditions (20) and (22) for a given 120574 le 1205722 If 119891 isin A and119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (37)

implies 10038161003816100381610038161003816arg1198651015840 (z)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (38)

Example 8 For 119891 isin A and 119865 = L119891 considering the specialcases 120574 = 1radic3 120574 = radic2 minus 1 and 120574 = 2 minus radic3 from the abovecorollary we obtain respectively the next implications

(1) If 2radic3 le 120572 le 43 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 1radic3) = 1205872 (120572 + 13) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(39)

(2) If 2(radic2 minus 1) le 120572 le 118 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572radic2 minus 1) = 1205872 (120572 + 14) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(40)

(3) If 2(2 minus radic3) le 120572 le 1712 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(41)

Theorem 9 For 119892 isin A suppose that the Libera transform119866 =L119892 isin NK(120573) 0 lt 120573 lt 1 Suppose that 1205720 is defined by (20)and for 0 lt 120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le fl (120572 120573)

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 (42)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161003816100381610038161003816arg 1198911015840 (119911)1198921015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U

(43)

implies 100381610038161003816100381610038161003816100381610038161003816arg 1198651015840 (119911)1198661015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120572 119911 isin U (44)

Proof Let 119901 be the function defined by

119901 (119911) = 1198651015840 (119911)1198661015840 (119911) 119911 isin U (45)

Similarly like in the proof of Theorem 6 we will prove thatthe function 119901 is analytic inU Supposing that there exists anumber 1199110 isin U such that 1198661015840(1199110) = 0 since 1198661015840(0) = 1 then1199110 isin U 0 Hence there exists a unique number 119899 isin N anda unique function119876 analytic inU with119876(1199110) = 0 such that

1198661015840 (119911) = (119911 minus 1199110)119899 119876 (119911) 119911 isin U (46)

From here we deduce that

11991111986610158401015840 (119911)1198661015840 (119911) = 119899119911119911 minus 1199110 + 1199111198761015840 (119911)119876 (119911) 119911 isin U 1199110 (47)

which implies that 1199110 is a simple pole for the function11991111986610158401015840(119911)1198661015840(119911) Now using the fact that 119866 = L119892 isin NK(120573)we obtain

0 = Re(1 + 11991111986610158401015840 (1199110)1198661015840 (1199110) )minus1 gt 120573 (48)

which contradicts the assumption 0 le 120573Consequently 1198661015840(119911) = 0 for all 119911 isin U and this implies

that the function 119901 is analytic inU with 119901(0) = 1From (45) we easily get

1198911015840 (119911)1198921015840 (119911) = 119901 (119911) + 120582 (119911) 1199111199011015840 (119911) (49)

where the function 120582 is given by (29)The assumption 119866 = L119892 isin NK(120573) is equivalent to

(1 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 ≺ 1 + (1 minus 2120573) 1199111 minus 119911 (50)

hence there exists a Schwarz function 119908 such that

(1 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 = 1 + (1 minus 2120573)119908 (119911)1 minus 119908 (119911) 119911 isin U (51)

By simple calculations we get

(2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 = 1 + (1 minus 2120573)119908 (119911)2 (1 minus 120573119908 (119911)) 119911 isin U (52)

that is

120582 (119911) = (2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 ≺ 1 + (1 minus 2120573) 1199112 (1 minus 120573119911) š 120601 (119911) (53)

Journal of Function Spaces 5

Since the circular transform 120601maps the unit diskU onto thediskΩ = 119908 isin C |119908 minus 120596| lt 119903

with 120596 = 1 + 21205732 (1 + 120573) 119903 = 12 (1 + 120573) (54)

from the subordination property (6) we deduce that thesubordination relation (53) is equivalent to

120582 (119911) = 119906 (119911) + 119894V (119911) isin Ω 119911 isin U (55)

With the above notation the condition (21) becomes

120572119906 (119911) minus 120572120574 |V (119911)| minus 120574 ge 0 119911 isin U (56)

We will prove that if 120574 satisfies the assumption (42) thatis 0 lt 120574 le then the subordination (53) implies that theinequality (56) holds that is

Ω sub Δ (57)

where

Δ = 119908 = 119906 + 119894V isin C 120572119906 minus 120572120574 |V| minus 120574 ge 0 (58)

Since the setsΩ and Δ are symmetric with respect to the realaxis the above inclusion is equivalent to

Ω sub Δ (59)

where

Ω = 119908 = 119906 + 119894V isin Ω V ge 0 Δ = 119908 = 119906 + 119894V isin Δ V ge 0 (60)

The line (119889) containing the half-line120572119906 minus 120572120574V minus 120574 = 0 V ge 0 (61)

which is a part of the boundary the set Δ has the equation120572119906 minus 120572120574V minus 120574 = 0 (62)

Thus the line (119889) intersects the real axis in the point 120574120572and contains the point minus119894120572 (see Figure 1) The smallest setΔ that includes the domain Ω is obtained for the case whenthe line (119889) containing the point minus119894120572 is tangent to the upperboundary of Ω which is the half-circle

(C) = 119908 = 119906 + 119894V isin C (119906 minus 120596)2 + V2 = 1199032 V ge 0 (63)

In this case as it is shown in the figure the line (119889) becomesthe tangent line (119889120591) to (C)

We will determine now the equation of the tangent line(119889120591) If we consider the family (119889119898) of all the lines containingthe point minus119894120572 with nonnegative angular coefficient that is

(119889119898) V = 119898119906 minus 1120572 119898 isin (0 +infin) (64)

u1

v

O

(d) (d120591)

Δ

Ω

120574

120572

120574

120572minus1

120572i

120573

1 + 120573

Figure 1 The inclusion Ω sub Δwe will solve the system

(119906 minus 120596)2 + V2 = 1199032 V ge 0V = 119898119906 minus 1120572 (65)

It follows that

1205722 (1 + 1198982) 1199062 minus 2120572 (120572120596 + 119898) 119906 + 1205722 (1205962 minus 1199032) + 1= 0 (66)

and the line is tangent to the half-circle if and only if thediscriminant of the above quadratic form is zero that is

1205722 (1205962 minus 1199032)1198982 minus 2120572120596119898 minus 12057221199032 + 1 = 0 (67)

This last equation has the biggest positive root

= 120596 + 119903radic(1205962 minus 1199032) 1205722 + 1120572 (1205962 minus 1199032) (68)

hence the tangent line to the half-circle (C) which containsthe point minus119894120572 has the equation

(119889120591) V = 119906 minus 1120572 (69)

The tangent (119889120591) intersects the real axis in the point 1(120572)hence we deduce that the inclusion (59) holds if and only if

0 lt 120574120572 le 1120572 (70)

that is

0 lt 120574 le 1119898 = 120572 (1205962 minus 1199032)120596 + 119903radic(1205962 minus 1199032) 1205722 + 1

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 = (120572 120573) (71)

Thus if the parameter 120574 satisfies the assumption (42)then the condition (21) of Lemma 5 is fulfilled and our resultfollows fromTheorem 6

6 Journal of Function Spaces

Putting 119892(119911) = 119911 in Theorem 9 for 120573 rarr 1minus we get thenext result

Corollary 10 Suppose that 1205720 is defined by (20) and for 0 lt120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le 21205723 + radic12057222 + 1 (72)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (73)

implies 10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (74)

Example 11 For 119891 isin A and 119865 = L119891 considering the specialcase 120574 = 2 minus radic3 from Corollary 10 we deduce the nextimplication

If

054552 ≃ minus168 + 108radic3 + 164radic2 minus 92radic647 le 120572le 1712

(75)

then 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(76)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Dr Badr S Alkahtani andDr Rubayyi Alqahtani extend theirsincere appropriations to the Deanship of Scientific Researchat King Saud University for funding this Profile ResearchGroup (PRG-1437-35)

References

[1] M Nunokawa S Owa J Nishiwaki K Kuroki and T HayamildquoDifferential subordination and argumental propertyrdquoComput-ers amp Mathematics with Applications vol 56 no 10 pp 2733ndash2736 2008

[2] V Ravichandran and S Sivaprasad Kumar ldquoArgument estimatefor starlike functions of reciprocal orderrdquo Southeast AsianBulletin of Mathematics vol 35 no 5 pp 837ndash843 2011

[3] C Pommerenke Univalent Functions Vanderhoeck and Ru-precht Gottingen Germany 1975

[4] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[5] M K Aouf J Dziok and J Sokol ldquoOn a subclass of stronglystarlike functionsrdquo Applied Mathematics Letters vol 24 no 1pp 27ndash32 2011

[6] N E Cho ldquoArgument estimates of certain meromorphic func-tionsrdquo Communications Korean Mathematical Society vol 15no 2 pp 263ndash274 2000

[7] N E Cho and J A Kim ldquoAngular estimations of certain analyticfunctionsrdquo Journal of the Korean Mathematical Society vol 34no 2 pp 427ndash436 1997

[8] N E Cho Y C Kim andHM Srivastava ldquoArgument estimatesfor a certain class of analytic functionsrdquo Complex VariablesTheory and Application vol 38 no 3 pp 277ndash287 1999

[9] N E Cho J Patel and G P Mohapatra ldquoArgument estimates ofcertain multivalent functions involving a linear operatorrdquo Inter-national Journal of Mathematics andMathematical Sciences vol31 no 11 pp 659ndash673 2002

[10] N E Cho and H M Srivastava ldquoArgument estimates of certainanalytic functions defined by a class of multiplier transforma-tionsrdquo Mathematical and Computer Modelling vol 37 no 1-2pp 39ndash49 2003

[11] N ECho S YWoo and SOwa ldquoArgument estimates of certainmeromorphic functionsrdquo Mathematica Balkanica New Seriesvol 15 no 3-4 pp 285ndash295 2001

[12] S Kanas andT Sugawa ldquoStrong starlikeness for a class of convexfunctionsrdquo Journal of Mathematical Analysis and Applicationsvol 336 no 2 pp 1005ndash1017 2007

[13] MNunokawa ldquoOn the order of strongly starlikeness of stronglyconvex functionsrdquo Japan Academy Proceedings Series A Math-ematical Sciences vol 69 no 7 pp 234ndash237 1993

[14] M Nunokawa S Owa H Saitoh and N N Pascu ldquoArgumentestimates for certain analytic functionsrdquo Proceedings of theJapan Academy Series A Mathematical Sciences vol 79 no 10pp 163ndash166 2003

[15] C Pommerenke ldquoOn close-to-convex analytic functionsrdquoTransactions of the American Mathematical Society vol 114 pp176ndash186 1965

[16] S Ponnusamy ldquoDifferential subordinations concerning starlikefunctionsrdquo Proceedings of the Indian Academy of SciencesmdashMathematical Sciences vol 104 no 2 pp 397ndash411 1994

[17] J K Prajapat and S P Goyal ldquoApplications of Srivastava-Attiyaoperator to the classes of strongly starlike and strongly convexfunctionsrdquo Journal of Mathematical Inequalities vol 3 no 1 pp129ndash137 2009

[18] V RavichandranM Darus andN Seenivasagan ldquoOn a criteriafor strong starlikenessrdquo Australian Journal of MathematicalAnalysis and Applications vol 2 no 1 article 6 12 pages 2005

[19] S Ponnusamy ldquoDifferential subordination and Bazilevic func-tionsrdquoProceedings of the IndianAcademy of SciencesMathemat-ical Sciences vol 105 no 2 pp 169ndash186 1995

[20] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces

Taking 119892(119911) = 119911 inTheorem 6 for the function 120582 definedby (29) we have 120582(119911) = 12 and we obtain the followingresult

Corollary 7 Suppose that the parameters 120572 and 120574 satisfy theconditions (20) and (22) for a given 120574 le 1205722 If 119891 isin A and119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (37)

implies 10038161003816100381610038161003816arg1198651015840 (z)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (38)

Example 8 For 119891 isin A and 119865 = L119891 considering the specialcases 120574 = 1radic3 120574 = radic2 minus 1 and 120574 = 2 minus radic3 from the abovecorollary we obtain respectively the next implications

(1) If 2radic3 le 120572 le 43 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 1radic3) = 1205872 (120572 + 13) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(39)

(2) If 2(radic2 minus 1) le 120572 le 118 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572radic2 minus 1) = 1205872 (120572 + 14) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(40)

(3) If 2(2 minus radic3) le 120572 le 1712 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(41)

Theorem 9 For 119892 isin A suppose that the Libera transform119866 =L119892 isin NK(120573) 0 lt 120573 lt 1 Suppose that 1205720 is defined by (20)and for 0 lt 120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le fl (120572 120573)

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 (42)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161003816100381610038161003816arg 1198911015840 (119911)1198921015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U

(43)

implies 100381610038161003816100381610038161003816100381610038161003816arg 1198651015840 (119911)1198661015840 (119911) 100381610038161003816100381610038161003816100381610038161003816 lt 1205872 120572 119911 isin U (44)

Proof Let 119901 be the function defined by

119901 (119911) = 1198651015840 (119911)1198661015840 (119911) 119911 isin U (45)

Similarly like in the proof of Theorem 6 we will prove thatthe function 119901 is analytic inU Supposing that there exists anumber 1199110 isin U such that 1198661015840(1199110) = 0 since 1198661015840(0) = 1 then1199110 isin U 0 Hence there exists a unique number 119899 isin N anda unique function119876 analytic inU with119876(1199110) = 0 such that

1198661015840 (119911) = (119911 minus 1199110)119899 119876 (119911) 119911 isin U (46)

From here we deduce that

11991111986610158401015840 (119911)1198661015840 (119911) = 119899119911119911 minus 1199110 + 1199111198761015840 (119911)119876 (119911) 119911 isin U 1199110 (47)

which implies that 1199110 is a simple pole for the function11991111986610158401015840(119911)1198661015840(119911) Now using the fact that 119866 = L119892 isin NK(120573)we obtain

0 = Re(1 + 11991111986610158401015840 (1199110)1198661015840 (1199110) )minus1 gt 120573 (48)

which contradicts the assumption 0 le 120573Consequently 1198661015840(119911) = 0 for all 119911 isin U and this implies

that the function 119901 is analytic inU with 119901(0) = 1From (45) we easily get

1198911015840 (119911)1198921015840 (119911) = 119901 (119911) + 120582 (119911) 1199111199011015840 (119911) (49)

where the function 120582 is given by (29)The assumption 119866 = L119892 isin NK(120573) is equivalent to

(1 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 ≺ 1 + (1 minus 2120573) 1199111 minus 119911 (50)

hence there exists a Schwarz function 119908 such that

(1 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 = 1 + (1 minus 2120573)119908 (119911)1 minus 119908 (119911) 119911 isin U (51)

By simple calculations we get

(2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 = 1 + (1 minus 2120573)119908 (119911)2 (1 minus 120573119908 (119911)) 119911 isin U (52)

that is

120582 (119911) = (2 + 11991111986610158401015840 (119911)1198661015840 (119911) )minus1 ≺ 1 + (1 minus 2120573) 1199112 (1 minus 120573119911) š 120601 (119911) (53)

Journal of Function Spaces 5

Since the circular transform 120601maps the unit diskU onto thediskΩ = 119908 isin C |119908 minus 120596| lt 119903

with 120596 = 1 + 21205732 (1 + 120573) 119903 = 12 (1 + 120573) (54)

from the subordination property (6) we deduce that thesubordination relation (53) is equivalent to

120582 (119911) = 119906 (119911) + 119894V (119911) isin Ω 119911 isin U (55)

With the above notation the condition (21) becomes

120572119906 (119911) minus 120572120574 |V (119911)| minus 120574 ge 0 119911 isin U (56)

We will prove that if 120574 satisfies the assumption (42) thatis 0 lt 120574 le then the subordination (53) implies that theinequality (56) holds that is

Ω sub Δ (57)

where

Δ = 119908 = 119906 + 119894V isin C 120572119906 minus 120572120574 |V| minus 120574 ge 0 (58)

Since the setsΩ and Δ are symmetric with respect to the realaxis the above inclusion is equivalent to

Ω sub Δ (59)

where

Ω = 119908 = 119906 + 119894V isin Ω V ge 0 Δ = 119908 = 119906 + 119894V isin Δ V ge 0 (60)

The line (119889) containing the half-line120572119906 minus 120572120574V minus 120574 = 0 V ge 0 (61)

which is a part of the boundary the set Δ has the equation120572119906 minus 120572120574V minus 120574 = 0 (62)

Thus the line (119889) intersects the real axis in the point 120574120572and contains the point minus119894120572 (see Figure 1) The smallest setΔ that includes the domain Ω is obtained for the case whenthe line (119889) containing the point minus119894120572 is tangent to the upperboundary of Ω which is the half-circle

(C) = 119908 = 119906 + 119894V isin C (119906 minus 120596)2 + V2 = 1199032 V ge 0 (63)

In this case as it is shown in the figure the line (119889) becomesthe tangent line (119889120591) to (C)

We will determine now the equation of the tangent line(119889120591) If we consider the family (119889119898) of all the lines containingthe point minus119894120572 with nonnegative angular coefficient that is

(119889119898) V = 119898119906 minus 1120572 119898 isin (0 +infin) (64)

u1

v

O

(d) (d120591)

Δ

Ω

120574

120572

120574

120572minus1

120572i

120573

1 + 120573

Figure 1 The inclusion Ω sub Δwe will solve the system

(119906 minus 120596)2 + V2 = 1199032 V ge 0V = 119898119906 minus 1120572 (65)

It follows that

1205722 (1 + 1198982) 1199062 minus 2120572 (120572120596 + 119898) 119906 + 1205722 (1205962 minus 1199032) + 1= 0 (66)

and the line is tangent to the half-circle if and only if thediscriminant of the above quadratic form is zero that is

1205722 (1205962 minus 1199032)1198982 minus 2120572120596119898 minus 12057221199032 + 1 = 0 (67)

This last equation has the biggest positive root

= 120596 + 119903radic(1205962 minus 1199032) 1205722 + 1120572 (1205962 minus 1199032) (68)

hence the tangent line to the half-circle (C) which containsthe point minus119894120572 has the equation

(119889120591) V = 119906 minus 1120572 (69)

The tangent (119889120591) intersects the real axis in the point 1(120572)hence we deduce that the inclusion (59) holds if and only if

0 lt 120574120572 le 1120572 (70)

that is

0 lt 120574 le 1119898 = 120572 (1205962 minus 1199032)120596 + 119903radic(1205962 minus 1199032) 1205722 + 1

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 = (120572 120573) (71)

Thus if the parameter 120574 satisfies the assumption (42)then the condition (21) of Lemma 5 is fulfilled and our resultfollows fromTheorem 6

6 Journal of Function Spaces

Putting 119892(119911) = 119911 in Theorem 9 for 120573 rarr 1minus we get thenext result

Corollary 10 Suppose that 1205720 is defined by (20) and for 0 lt120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le 21205723 + radic12057222 + 1 (72)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (73)

implies 10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (74)

Example 11 For 119891 isin A and 119865 = L119891 considering the specialcase 120574 = 2 minus radic3 from Corollary 10 we deduce the nextimplication

If

054552 ≃ minus168 + 108radic3 + 164radic2 minus 92radic647 le 120572le 1712

(75)

then 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(76)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Dr Badr S Alkahtani andDr Rubayyi Alqahtani extend theirsincere appropriations to the Deanship of Scientific Researchat King Saud University for funding this Profile ResearchGroup (PRG-1437-35)

References

[1] M Nunokawa S Owa J Nishiwaki K Kuroki and T HayamildquoDifferential subordination and argumental propertyrdquoComput-ers amp Mathematics with Applications vol 56 no 10 pp 2733ndash2736 2008

[2] V Ravichandran and S Sivaprasad Kumar ldquoArgument estimatefor starlike functions of reciprocal orderrdquo Southeast AsianBulletin of Mathematics vol 35 no 5 pp 837ndash843 2011

[3] C Pommerenke Univalent Functions Vanderhoeck and Ru-precht Gottingen Germany 1975

[4] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[5] M K Aouf J Dziok and J Sokol ldquoOn a subclass of stronglystarlike functionsrdquo Applied Mathematics Letters vol 24 no 1pp 27ndash32 2011

[6] N E Cho ldquoArgument estimates of certain meromorphic func-tionsrdquo Communications Korean Mathematical Society vol 15no 2 pp 263ndash274 2000

[7] N E Cho and J A Kim ldquoAngular estimations of certain analyticfunctionsrdquo Journal of the Korean Mathematical Society vol 34no 2 pp 427ndash436 1997

[8] N E Cho Y C Kim andHM Srivastava ldquoArgument estimatesfor a certain class of analytic functionsrdquo Complex VariablesTheory and Application vol 38 no 3 pp 277ndash287 1999

[9] N E Cho J Patel and G P Mohapatra ldquoArgument estimates ofcertain multivalent functions involving a linear operatorrdquo Inter-national Journal of Mathematics andMathematical Sciences vol31 no 11 pp 659ndash673 2002

[10] N E Cho and H M Srivastava ldquoArgument estimates of certainanalytic functions defined by a class of multiplier transforma-tionsrdquo Mathematical and Computer Modelling vol 37 no 1-2pp 39ndash49 2003

[11] N ECho S YWoo and SOwa ldquoArgument estimates of certainmeromorphic functionsrdquo Mathematica Balkanica New Seriesvol 15 no 3-4 pp 285ndash295 2001

[12] S Kanas andT Sugawa ldquoStrong starlikeness for a class of convexfunctionsrdquo Journal of Mathematical Analysis and Applicationsvol 336 no 2 pp 1005ndash1017 2007

[13] MNunokawa ldquoOn the order of strongly starlikeness of stronglyconvex functionsrdquo Japan Academy Proceedings Series A Math-ematical Sciences vol 69 no 7 pp 234ndash237 1993

[14] M Nunokawa S Owa H Saitoh and N N Pascu ldquoArgumentestimates for certain analytic functionsrdquo Proceedings of theJapan Academy Series A Mathematical Sciences vol 79 no 10pp 163ndash166 2003

[15] C Pommerenke ldquoOn close-to-convex analytic functionsrdquoTransactions of the American Mathematical Society vol 114 pp176ndash186 1965

[16] S Ponnusamy ldquoDifferential subordinations concerning starlikefunctionsrdquo Proceedings of the Indian Academy of SciencesmdashMathematical Sciences vol 104 no 2 pp 397ndash411 1994

[17] J K Prajapat and S P Goyal ldquoApplications of Srivastava-Attiyaoperator to the classes of strongly starlike and strongly convexfunctionsrdquo Journal of Mathematical Inequalities vol 3 no 1 pp129ndash137 2009

[18] V RavichandranM Darus andN Seenivasagan ldquoOn a criteriafor strong starlikenessrdquo Australian Journal of MathematicalAnalysis and Applications vol 2 no 1 article 6 12 pages 2005

[19] S Ponnusamy ldquoDifferential subordination and Bazilevic func-tionsrdquoProceedings of the IndianAcademy of SciencesMathemat-ical Sciences vol 105 no 2 pp 169ndash186 1995

[20] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces 5

Since the circular transform 120601maps the unit diskU onto thediskΩ = 119908 isin C |119908 minus 120596| lt 119903

with 120596 = 1 + 21205732 (1 + 120573) 119903 = 12 (1 + 120573) (54)

from the subordination property (6) we deduce that thesubordination relation (53) is equivalent to

120582 (119911) = 119906 (119911) + 119894V (119911) isin Ω 119911 isin U (55)

With the above notation the condition (21) becomes

120572119906 (119911) minus 120572120574 |V (119911)| minus 120574 ge 0 119911 isin U (56)

We will prove that if 120574 satisfies the assumption (42) thatis 0 lt 120574 le then the subordination (53) implies that theinequality (56) holds that is

Ω sub Δ (57)

where

Δ = 119908 = 119906 + 119894V isin C 120572119906 minus 120572120574 |V| minus 120574 ge 0 (58)

Since the setsΩ and Δ are symmetric with respect to the realaxis the above inclusion is equivalent to

Ω sub Δ (59)

where

Ω = 119908 = 119906 + 119894V isin Ω V ge 0 Δ = 119908 = 119906 + 119894V isin Δ V ge 0 (60)

The line (119889) containing the half-line120572119906 minus 120572120574V minus 120574 = 0 V ge 0 (61)

which is a part of the boundary the set Δ has the equation120572119906 minus 120572120574V minus 120574 = 0 (62)

Thus the line (119889) intersects the real axis in the point 120574120572and contains the point minus119894120572 (see Figure 1) The smallest setΔ that includes the domain Ω is obtained for the case whenthe line (119889) containing the point minus119894120572 is tangent to the upperboundary of Ω which is the half-circle

(C) = 119908 = 119906 + 119894V isin C (119906 minus 120596)2 + V2 = 1199032 V ge 0 (63)

In this case as it is shown in the figure the line (119889) becomesthe tangent line (119889120591) to (C)

We will determine now the equation of the tangent line(119889120591) If we consider the family (119889119898) of all the lines containingthe point minus119894120572 with nonnegative angular coefficient that is

(119889119898) V = 119898119906 minus 1120572 119898 isin (0 +infin) (64)

u1

v

O

(d) (d120591)

Δ

Ω

120574

120572

120574

120572minus1

120572i

120573

1 + 120573

Figure 1 The inclusion Ω sub Δwe will solve the system

(119906 minus 120596)2 + V2 = 1199032 V ge 0V = 119898119906 minus 1120572 (65)

It follows that

1205722 (1 + 1198982) 1199062 minus 2120572 (120572120596 + 119898) 119906 + 1205722 (1205962 minus 1199032) + 1= 0 (66)

and the line is tangent to the half-circle if and only if thediscriminant of the above quadratic form is zero that is

1205722 (1205962 minus 1199032)1198982 minus 2120572120596119898 minus 12057221199032 + 1 = 0 (67)

This last equation has the biggest positive root

= 120596 + 119903radic(1205962 minus 1199032) 1205722 + 1120572 (1205962 minus 1199032) (68)

hence the tangent line to the half-circle (C) which containsthe point minus119894120572 has the equation

(119889120591) V = 119906 minus 1120572 (69)

The tangent (119889120591) intersects the real axis in the point 1(120572)hence we deduce that the inclusion (59) holds if and only if

0 lt 120574120572 le 1120572 (70)

that is

0 lt 120574 le 1119898 = 120572 (1205962 minus 1199032)120596 + 119903radic(1205962 minus 1199032) 1205722 + 1

= 21205721205731 + 2120573 + radic(120573 (1 + 120573)) 1205722 + 1 = (120572 120573) (71)

Thus if the parameter 120574 satisfies the assumption (42)then the condition (21) of Lemma 5 is fulfilled and our resultfollows fromTheorem 6

6 Journal of Function Spaces

Putting 119892(119911) = 119911 in Theorem 9 for 120573 rarr 1minus we get thenext result

Corollary 10 Suppose that 1205720 is defined by (20) and for 0 lt120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le 21205723 + radic12057222 + 1 (72)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (73)

implies 10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (74)

Example 11 For 119891 isin A and 119865 = L119891 considering the specialcase 120574 = 2 minus radic3 from Corollary 10 we deduce the nextimplication

If

054552 ≃ minus168 + 108radic3 + 164radic2 minus 92radic647 le 120572le 1712

(75)

then 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(76)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Dr Badr S Alkahtani andDr Rubayyi Alqahtani extend theirsincere appropriations to the Deanship of Scientific Researchat King Saud University for funding this Profile ResearchGroup (PRG-1437-35)

References

[1] M Nunokawa S Owa J Nishiwaki K Kuroki and T HayamildquoDifferential subordination and argumental propertyrdquoComput-ers amp Mathematics with Applications vol 56 no 10 pp 2733ndash2736 2008

[2] V Ravichandran and S Sivaprasad Kumar ldquoArgument estimatefor starlike functions of reciprocal orderrdquo Southeast AsianBulletin of Mathematics vol 35 no 5 pp 837ndash843 2011

[3] C Pommerenke Univalent Functions Vanderhoeck and Ru-precht Gottingen Germany 1975

[4] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[5] M K Aouf J Dziok and J Sokol ldquoOn a subclass of stronglystarlike functionsrdquo Applied Mathematics Letters vol 24 no 1pp 27ndash32 2011

[6] N E Cho ldquoArgument estimates of certain meromorphic func-tionsrdquo Communications Korean Mathematical Society vol 15no 2 pp 263ndash274 2000

[7] N E Cho and J A Kim ldquoAngular estimations of certain analyticfunctionsrdquo Journal of the Korean Mathematical Society vol 34no 2 pp 427ndash436 1997

[8] N E Cho Y C Kim andHM Srivastava ldquoArgument estimatesfor a certain class of analytic functionsrdquo Complex VariablesTheory and Application vol 38 no 3 pp 277ndash287 1999

[9] N E Cho J Patel and G P Mohapatra ldquoArgument estimates ofcertain multivalent functions involving a linear operatorrdquo Inter-national Journal of Mathematics andMathematical Sciences vol31 no 11 pp 659ndash673 2002

[10] N E Cho and H M Srivastava ldquoArgument estimates of certainanalytic functions defined by a class of multiplier transforma-tionsrdquo Mathematical and Computer Modelling vol 37 no 1-2pp 39ndash49 2003

[11] N ECho S YWoo and SOwa ldquoArgument estimates of certainmeromorphic functionsrdquo Mathematica Balkanica New Seriesvol 15 no 3-4 pp 285ndash295 2001

[12] S Kanas andT Sugawa ldquoStrong starlikeness for a class of convexfunctionsrdquo Journal of Mathematical Analysis and Applicationsvol 336 no 2 pp 1005ndash1017 2007

[13] MNunokawa ldquoOn the order of strongly starlikeness of stronglyconvex functionsrdquo Japan Academy Proceedings Series A Math-ematical Sciences vol 69 no 7 pp 234ndash237 1993

[14] M Nunokawa S Owa H Saitoh and N N Pascu ldquoArgumentestimates for certain analytic functionsrdquo Proceedings of theJapan Academy Series A Mathematical Sciences vol 79 no 10pp 163ndash166 2003

[15] C Pommerenke ldquoOn close-to-convex analytic functionsrdquoTransactions of the American Mathematical Society vol 114 pp176ndash186 1965

[16] S Ponnusamy ldquoDifferential subordinations concerning starlikefunctionsrdquo Proceedings of the Indian Academy of SciencesmdashMathematical Sciences vol 104 no 2 pp 397ndash411 1994

[17] J K Prajapat and S P Goyal ldquoApplications of Srivastava-Attiyaoperator to the classes of strongly starlike and strongly convexfunctionsrdquo Journal of Mathematical Inequalities vol 3 no 1 pp129ndash137 2009

[18] V RavichandranM Darus andN Seenivasagan ldquoOn a criteriafor strong starlikenessrdquo Australian Journal of MathematicalAnalysis and Applications vol 2 no 1 article 6 12 pages 2005

[19] S Ponnusamy ldquoDifferential subordination and Bazilevic func-tionsrdquoProceedings of the IndianAcademy of SciencesMathemat-ical Sciences vol 105 no 2 pp 169ndash186 1995

[20] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces

Putting 119892(119911) = 119911 in Theorem 9 for 120573 rarr 1minus we get thenext result

Corollary 10 Suppose that 1205720 is defined by (20) and for 0 lt120572 le 1205720 let 120574 satisfy the condition0 lt 120574 le 21205723 + radic12057222 + 1 (72)

If 119891 isin A and 119865 = L119891 then100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 120574) = 1205872 (120572 + 2120587 tanminus1120574) 119911 isin U (73)

implies 10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U (74)

Example 11 For 119891 isin A and 119865 = L119891 considering the specialcase 120574 = 2 minus radic3 from Corollary 10 we deduce the nextimplication

If

054552 ≃ minus168 + 108radic3 + 164radic2 minus 92radic647 le 120572le 1712

(75)

then 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt 1205872 120578 (120572 2 minus radic3) = 1205872 (120572 + 16) 119911 isin U

dArr10038161003816100381610038161003816arg1198651015840 (119911)10038161003816100381610038161003816 lt 1205872 120572 119911 isin U(76)

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Dr Badr S Alkahtani andDr Rubayyi Alqahtani extend theirsincere appropriations to the Deanship of Scientific Researchat King Saud University for funding this Profile ResearchGroup (PRG-1437-35)

References

[1] M Nunokawa S Owa J Nishiwaki K Kuroki and T HayamildquoDifferential subordination and argumental propertyrdquoComput-ers amp Mathematics with Applications vol 56 no 10 pp 2733ndash2736 2008

[2] V Ravichandran and S Sivaprasad Kumar ldquoArgument estimatefor starlike functions of reciprocal orderrdquo Southeast AsianBulletin of Mathematics vol 35 no 5 pp 837ndash843 2011

[3] C Pommerenke Univalent Functions Vanderhoeck and Ru-precht Gottingen Germany 1975

[4] S SMiller andP TMocanuDifferential SubordinationsTheoryand Applications vol 225 ofMonographs and Textbooks in Pureand Applied Mathematics Marcel Dekker New York NY USA2000

[5] M K Aouf J Dziok and J Sokol ldquoOn a subclass of stronglystarlike functionsrdquo Applied Mathematics Letters vol 24 no 1pp 27ndash32 2011

[6] N E Cho ldquoArgument estimates of certain meromorphic func-tionsrdquo Communications Korean Mathematical Society vol 15no 2 pp 263ndash274 2000

[7] N E Cho and J A Kim ldquoAngular estimations of certain analyticfunctionsrdquo Journal of the Korean Mathematical Society vol 34no 2 pp 427ndash436 1997

[8] N E Cho Y C Kim andHM Srivastava ldquoArgument estimatesfor a certain class of analytic functionsrdquo Complex VariablesTheory and Application vol 38 no 3 pp 277ndash287 1999

[9] N E Cho J Patel and G P Mohapatra ldquoArgument estimates ofcertain multivalent functions involving a linear operatorrdquo Inter-national Journal of Mathematics andMathematical Sciences vol31 no 11 pp 659ndash673 2002

[10] N E Cho and H M Srivastava ldquoArgument estimates of certainanalytic functions defined by a class of multiplier transforma-tionsrdquo Mathematical and Computer Modelling vol 37 no 1-2pp 39ndash49 2003

[11] N ECho S YWoo and SOwa ldquoArgument estimates of certainmeromorphic functionsrdquo Mathematica Balkanica New Seriesvol 15 no 3-4 pp 285ndash295 2001

[12] S Kanas andT Sugawa ldquoStrong starlikeness for a class of convexfunctionsrdquo Journal of Mathematical Analysis and Applicationsvol 336 no 2 pp 1005ndash1017 2007

[13] MNunokawa ldquoOn the order of strongly starlikeness of stronglyconvex functionsrdquo Japan Academy Proceedings Series A Math-ematical Sciences vol 69 no 7 pp 234ndash237 1993

[14] M Nunokawa S Owa H Saitoh and N N Pascu ldquoArgumentestimates for certain analytic functionsrdquo Proceedings of theJapan Academy Series A Mathematical Sciences vol 79 no 10pp 163ndash166 2003

[15] C Pommerenke ldquoOn close-to-convex analytic functionsrdquoTransactions of the American Mathematical Society vol 114 pp176ndash186 1965

[16] S Ponnusamy ldquoDifferential subordinations concerning starlikefunctionsrdquo Proceedings of the Indian Academy of SciencesmdashMathematical Sciences vol 104 no 2 pp 397ndash411 1994

[17] J K Prajapat and S P Goyal ldquoApplications of Srivastava-Attiyaoperator to the classes of strongly starlike and strongly convexfunctionsrdquo Journal of Mathematical Inequalities vol 3 no 1 pp129ndash137 2009

[18] V RavichandranM Darus andN Seenivasagan ldquoOn a criteriafor strong starlikenessrdquo Australian Journal of MathematicalAnalysis and Applications vol 2 no 1 article 6 12 pages 2005

[19] S Ponnusamy ldquoDifferential subordination and Bazilevic func-tionsrdquoProceedings of the IndianAcademy of SciencesMathemat-ical Sciences vol 105 no 2 pp 169ndash186 1995

[20] R J Libera ldquoSome classes of regular univalent functionsrdquo Pro-ceedings of the American Mathematical Society vol 16 pp 755ndash758 1965

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of