A Study of Third and Fourth Hankel Determinant Problem for

Research ArticleA Study of Third and Fourth Hankel Determinant Problem for aParticular Class of Bounded Turning Functions

Gaganpreet Kaur1 Gurmeet Singh2 Muhammad Arif 3 Ronnason Chinram 4

and Javed Iqbal3

1Department of Mathematics Punjabi University Patiala 147001 Punjab India2Department of Mathematics Khalsa College Patiala 147002 Punjab India3Department of Mathematics Abdul Wali khan University Mardan Mardan 23200 Pakistan4Division of Computational Science Faculty of Science Prince of Songkla University Hat Yai Songkhla 90110 -ailand

Correspondence should be addressed to Muhammad Arif marifmathsawkumedupk

Received 29 November 2020 Revised 15 March 2021 Accepted 3 April 2021 Published 19 April 2021

Academic Editor Erhan Set

Copyright copy 2021 Gaganpreet Kaur et al +is 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 isproperly cited

In this present paper a new generalized class Rpq from the family of function with bounded turning was introduced by using(p q)- derivative operator Our aim for this class is to find out the upper bound of third- and fourth-order Hankel determinantMoreover the upper bounds for two-fold and three-fold symmetric functions for this class are also obtained

1 Introduction and Motivation

In order to better explain the terminology included in ourkey observations some of the essential relevant literatures ongeometric function theory need to be provided and dis-cussed here We start with symbol A which represents theclass of holomorphic functions in the region of open unitdisc U z |z|lt 1 and satisfy the relationshipg(0) gprime(0) minus 1 0 for g isin A +at is if g isin A then it hasthe following Taylor series form

g(z) z + 1113944infin

λ2δλz

λ (z isin U) (1)

Also letΨ sub A represent all univalent functions inU Nextwe are going to define the most useful class of geometricfunction theory known as the classP of Caratheodory functionsand is defines as a holomorphic function h belongs to P if itsatisfies Re(h(z))gt 0 along with the series expansion

h(z) 1 + 1113944infin

λ1dλz

λ (z isin U) (2)

Using the Caratheodory functions family we considerthe following basics subclasses of Ψ as

Slowast

g(z) isin Azgprime(z)

g(z)isin P (z isin U)1113896 1113897

K g(z) isin Azgprime(z)( 1113857prime

gprime(z)isin P (z isin U)1113896 1113897

R g(z) isin A gprime(z) isin P (z isin U)1113864 1113865

(3)

+e investigation of q-calculus (q stands for quantum)fascinated and inspired many scholars due its use invarious areas of the quantitative sciences Jackson [1 2]was among the key contributors of all the scientists whointroduced and developed the q-calculus theory Just likeq-calculus was used in other mathematical sciences theformulations of this idea are commonly used to examinethe existence of various structures of function theory+ough the link between certain geometric nature of theanalytic function and the q-derivative operator wasestablished by the authors in [3] but for the usage ofq-calculus in function theory a solid and comprehensive

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 6687805 8 pageshttpsdoiorg10115520216687805

foundation is given in [4] by Srivastava After this de-velopment many researchers introduced and studiedsome useful operators in q-analog with the applications ofconvolution concepts For example Kanas and Raducanu[5] established the q-differential operator and then ex-amined the behavior of this operator in function theory+is operator was generalized further for multivalentanalytic functions by Arif et al [6] Analogous to q-dif-ferential operator Arif et al and Khan et al contributedthe integral operators for analytic and multivalent func-tions in [7 8] respectively Similarly in the article [9] theauthors developed and analyzed the operator in q-analogfor meromorphic functions Also see the survey typearticle [10] on quantum calculus and their applicationsWith the use of these operators many researchers werecontributed some good articles in this direction in thefield of geometric function theory see [11 12] +esubstitution of q by qp in q-calculus has given us(p q)-calculus which is the extension of q-calculusChakrabarti and Jagannathan [13] considered(p q)-integer

For 0lt qltple 1 and zne 0 the (p q)-derivative operatorDpqg of the function g(z) isin A is defined as

Dpqg(z) g(pz) minus g(qz)

(p minus q)z (4)

From the use (1) and (4) the following formulae areeasily obtained

Dpq g1(z) + g2(z)( 1113857 Dpqg1(z) + Dpqg2(z) (5)

and for a constant c

Dpq(cg(z)) cDpqg(z) (6)

Here we also note that Dpqg(z) gprime(z) when we takep 1 and q⟶ 1minus

limq⟶1minus

Dpq(g(z)) limq⟶1minus

g(z) minus g(qz)

(1 minus q)z gprime(z) (7)

Further by using (1) and (4) we have

Dpq(g(z)) 1 + 1113944infin

λ2[λ]pqδλz

λminus 1 (8)

with

[λ]pq pλ

minus qλ

p minus q p

λminus 1 1 + 1113944λminus1

l1

q

p1113888 1113889

l

⎛⎝ ⎞⎠ (9)

Let us define the class Rpq which consists all thefunctions g isin A satisfying

Re Dpq(g(z))1113872 1113873gt 0 z isin U (10)

+e Hankel determinant HDnλ(g) withn λ isin N 1 2 and δ1 1 for a function g isin Ψ of theseries form (1) was given by Pommerenke [14 15] as

HDnλ(g)

δλ δλ+1 middot middot middot δλ+nminus1

δλ+1 δλ+2 middot middot middot δλ+n

⋮ ⋮ ⋮ ⋮

δλ+nminus1 δλ+n middot middot middot δλ+2nminus2

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(11)

In particular the following determinants are known asthe first- second- and third-order Hankel determinantsrespectively

HD21(g) 1 δ2δ2 δ3

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 δ3 minus δ22

HD22(g) δ2 δ3δ3 δ4

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 δ2δ4 minus δ23

HD31(g)

1 δ2 δ3δ2 δ3 δ4δ3 δ4 δ5

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

δ3 δ2δ4 minus δ231113872 1113873 minus δ4 δ4 minus δ2δ3( 1113857

+ δ5 δ3 minus δ221113872 1113873

(12)

+ere are comparatively few observations in literature inrelation to the Hankel determinant for the function g whichbelongs to the general family Ψ For the function g isin Ψ thebest established sharp inequality is |HD2λg|le μ

λ

radic where μ

is the absolute constant which is due to Hayman [16]Further for the same class Ψ it was obtained in [17] that

HD22(g)1113868111386811138681113868

1113868111386811138681113868le μ1 for 1le μ1 le113

HD31(g)1113868111386811138681113868

1113868111386811138681113868le μ2 for49le μ2 le

32 +285

radic

15

(13)

In a given family of functions the problem of calculatingthe bounds probably sharp of Hankel determinantsattracted the minds of several mathematicians For examplethe sharp bound of |HD22(g)| for the subfamilies K Slowastand R (family of bounded turning functions) of the set Ψwas calculated by Janteng et al [18 19] +ese estimates are

HD22(g)1113868111386811138681113868

1113868111386811138681113868le

18 forf isinK

1 forf isin Slowast

49 forf isinR

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

For the following two families Slowast(β) (0le βlt 1) of star-like functions of order β and for SSlowast(β) (0lt βle 1) ofstrongly star-like functions of order β the authors computedin [20 21] that |HD22(g)| is bounded by (1 minus β)2 and β2respectively +e exact bound for the family of Ma-Mindastar-like functions was measured in [22] see also [23] Formore work on |HD22(g)| see references [24ndash28]

It is quite clear from the formulae given in (12) that thecalculation of |HD31(g)| is far more challenging compared

2 Mathematical Problems in Engineering

with finding the bound of |HD22(g)| Babalola was the firstmathematician who investigated the bounds of thirdorderHankel determinant for the families of K Slowast and R in anarticle [29] published in 2010 Using the same approachlater several authors [30ndash34] published their articles re-garding |HD31(g)| for certain subfamilies of analytic andunivalent functions After this study Zaprawa [35] im-proved the findings of Babalola in 2017 by applying a newmethodology He obtained the following bounds

HD31(g)1113868111386811138681113868

1113868111386811138681113868le

49540

forg isinK

1 forg isin Slowast

4160

forg isinR

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

He argued that such limits are indeed not the best Laterin 2018 Kwon et al [36] strengthened the Zaprawarsquos resultfor g isin Slowast and showed that |HD31(g)|le (89) and thisbound was further improved by Zaprawa et al [37] in 2021+ey got |HD31(g)|le (59) for g isin Slowast Recently in 2018Kowalczyk et al [38] and Lecko et al [39] succeeded infinding the sharp bounds of |HD31(g)| for the families Kand Slowast(12) respectively where Slowast(12) indicate the star-like functions family of order 12 +ese results are given as

HD31(g)1113868111386811138681113868

1113868111386811138681113868le

4135

for g isinK

19 for g isin Slowast

12

1113874 1113875

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)

+e estimation of fourth Hankel determinantHD41(g)

for the bounded turning functions has been obtained by Arifet al [40] and they proved the following bounds for g isinR

HD41(g)1113868111386811138681113868

1113868111386811138681113868le 078050 (17)

After that Kaur and Singh [41] proved fourth Hankeldeterminant for bounded turning function of order α Formore contributions see [42ndash46] Recently Srivastava et al[47] consider a family of normalized analytic functions withbounded turnings in the open unit disk which are connectedwith the cardioid domains and they obtained the estimates offourth Hankel determinant

2 A Set of Results

In order to investigate HD41(g) we need the followingresults

Lemma 1 (see [48]) If h(z) isin P having the form (2) then

dλ1113868111386811138681113868

1113868111386811138681113868|2 for λ isin N (18)

dλ+k minus μdλdk

11138681113868111386811138681113868111386811138681113868le 2 for 0le μle 1 (19)

dpdq minus drds

11138681113868111386811138681113868

11138681113868111386811138681113868le 4 forp + q r + s (20)

Theorem 1 (see [40]) Let g(z) z + 1113936infinλ2 δλz

λ isin Slowast andfor real μ

δ22 δ3 minus μδ221113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868

4(3 minus 4μ) for μle58

12(2μ minus 1)

for μ isin5834

1113876 1113877

14(1 minus μ)

for μ isin3478

1113876 1113877

4(4μ minus 3) for μge78

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

3 Bounds of Third Hankel Determinant

+e third Hankel determinant HD31(g) is a polynomial offour variables as

HD31(g) δ3δ5 minus δ241113872 1113873 + δ2 δ3δ4 minus δ2δ5( 1113857 + δ3 δ2δ4 minus δ231113872 1113873

(22)

In order to solve HD31(g) we need to know thecorrespondence between g and h isin P

g(z) isinRpqhArrDpqg(z) isin P (23)

+us

1 + 1113944infin

λ2[λ]pqδλz

λminus 1⎛⎝ ⎞⎠ 1 + 1113944infin

λ1dλz

λ⎛⎝ ⎞⎠ (24)

By simplifying we yield

δλ dλminus1

[λ]pq

(25)

Now using the above coefficients in (22) we obtain

HD31(g) 1

[3]pq[5]pq

d2d4 minus1

[4]2pq

d23 +

2[2]pq[3]pq[4]pq

d1d2d3 minus1

[2]2pq[5]pq

d21d4 minus

1[3]

3pq

d32 (26)

Mathematical Problems in Engineering 3

Rearranging the above terms

HD31(g) 1

[2]2pq[5]pq

d4 d2 minus d211113872 1113873 minus

1[4]

2pq

d3 d3 minus d1d2( 1113857 +1

[3]3pq

d2 d4 minus d221113872 1113873

minus2

[2]pq[3]pq[4]pq

minus1

[4]2pq

⎛⎝ ⎞⎠d2 d4 minus d1d3( 1113857

+1

[3]pq[5]pq

minus1

[2]2pq[5]pq

minus1

[3]3pq

+2

[2]pq[3]pq[4]pq

minus1

[4]2pq

⎛⎝ ⎞⎠d2d4

(27)

Using triangular inequality and the results (18) and (19)of lemma of Section 2 we get

|HD31(g)|le 44

[2]pq[3]pq[4]pq

+1

[3]pq[5]pq

minus1

[4]2pq

⎛⎝ ⎞⎠

(28)

Remark 1 As we know by (7) if p 1 and q⟶ 1minus thenthe above result (28) coincides with Zaprawa [35]

4 Bounds of Fourth Hankel Determinant

Firstly HD41(g) is the fourth Hankel determinant of theform (11) with six coefficients which can be written in theform

HD41(g) δ7HD31(g) minus δ6Λ1 + δ5Λ2 minus δ4Λ3 (29)

where Λ1Λ2 and Λ3 are third-order determinants given as

Λ1 δ3δ6 minus δ4δ5( 1113857 minus δ2 δ2δ6 minus δ3δ5( 1113857 + δ4 δ2δ4 minus δ231113872 1113873

(30)

Λ2 δ4δ6 minus δ251113872 1113873 minus δ2 δ3δ6 minus δ4δ5( 1113857 + δ3 δ3δ5 minus δ241113872 1113873 (31)

Λ3 δ2 δ4δ6 minus δ251113872 1113873 minus δ3 δ3δ6 minus δ4δ5( 1113857 + δ4 δ3δ5 minus δ241113872 1113873

(32)

Theorem 2 If g isinRpq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le 81

[3]pq[7]pq

σ1 +1

[4]2pq

σ2 +1

[5]2pq

σ3 +1

[6]pq

σ4⎛⎝ ⎞⎠

(33)

where

σ1 4

[2]pq[4]pq

+1

[5]pq

1113888 1113889

σ2 2

[2]pq[6]pq

minus1

[7]pq

+4

[3]pq[5]pq

1113888 1113889

σ3 2

[2]pq[4]pq

+2

[3]2pq

minus1

[5]pq

⎛⎝ ⎞⎠

σ4 2

[3]2pq[4]pq

+2

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠

(34)

Proof By using definition (10) we have g(z) isinRpq +usDpqg(z) h(z) isin P

1 + 1113944infin

λ2[λ]pqδλz

λminus 1⎛⎝ ⎞⎠ 1 + 1113944infin

λ1dλz

λ⎛⎝ ⎞⎠ (35)

+erefore

[λ]pqδλ dλminus1 (36)

Substituting (36) in (30) and (31) and in (32) yields that

Λ1 1

[3]pq[6]pq

d2d5 minus1

[4]pq[5]pq

d3d4 minus1

[2]2pq[6]pq

d21d5 +

1[2]pq[3]pq[5]pq

d1d2d4 +1

[2]pq[4]2pq

d1d23

minus1

[3]2pq[4]pq

d22d3

Λ2 1

[4]pq[6]pq

d3d5 minus1

[5]2pq

d24 minus

1[2]pq[3]pq[6]pq

d1d2d5 +1

[2]pq[4]pq[5]pq

d1d3d4 +1

[3]2pq[5]pq

d22d4

minus1

[3]pq[4]2pq

d2d23

Λ3 1

[2]pq[4]pq[6]pq

d1d3d5 minus1

[2]pq[5]2pq

d1d24 minus

1[3]

2pq[6]pq

d22d5 +

2[3]pq[4]pq[5]pq

d2d3d4 minus1

[4]3pq

d33

(37)


Now rewrite the above equations as follows

Λ1 1

[2]2pq[6]pq

d5 d2 minus d211113872 1113873 +

1[3]

2pq[4]pq

d3 d4 minus d221113872 1113873 minus

1[2]pq[4]

2pq

d3 d4 minus d1d3( 1113857

minus1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

⎛⎝ ⎞⎠d4 d3 minus d1d2( 1113857

+1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

minus1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2 d5 minus d1d4( 1113857

+1

[6]pq[3]pq

minus1

[6]pq[2]2pq

minus1

[4]pq[5]pq

minus1

[3]2pq[4]pq

+1

[2]pq[4]2pq

+1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2d5

Λ2 1

[2]pq[3]pq[6]pq

d5 d3 minus d1d2( 1113857 minus1

[3]2pq[5]pq

d4 d4 minus d221113872 1113873 +

1[3]pq[4]

2pq

d3 d5 minus d2d3( 1113857

minus1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3 d5 minus d1d4( 1113857 minus1

[5]2pq

minus1

[3]2pq[5]pq

⎛⎝ ⎞⎠d4 d4 minus d1d3( 1113857

+1

[4]pq[6]pq

minus1

[2]pq[3]pq[6]pq

minus1

[3]pq[4]2pq

+1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3d5

Λ3 1

[3]2pq[6]pq

d5 d4 minus d221113872 1113873 minus

1[2]pq[4]pq[6]pq

d5 d4 minus d1d3( 1113857 +1

[4]3pq

d3 d6 minus d231113872 1113873

minus1

[4]3pq

d3 d6 minus d2d4( 1113857 +1

[2]pq[5]2pq


[3]pq[4]pq[5]pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4 d5 minus d2d3( 1113857

+1

[2]pq[4]pq[6]pq

+2

[3]pq[4]pq[5]pq

minus1

[3]2pq[6]pq

minus1

[2]pq[5]2pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4d5

(38)

Using the triangular inequality with the inequalities (18)and (19) of lemma on the above equations we obtain

Λ11113868111386811138681113868

1113868111386811138681113868le 42

[3]2pq[4]pq

+1

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠ (39)

Λ21113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[5]pq

+2

[3]2pq[5]pq

+1

[4]pq[6]pq

minus1

[5]2pq

⎛⎝ ⎞⎠

(40)

Λ31113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[6]pq

+4

[3]pq[4]pq[5]pq

1113888 1113889 (41)

Now using the values (39)ndash(41) and (28) along with theinequality |δλ|le (2[λ]pq) in (29) we get our desiredresult

5 BoundsofHD41(g) forTwo-FoldandThree-Fold Symmetric Functions

n-Fold symmetric function consists all those functions g

which satisfy the following condition

g(εz) εg(z) forallz isin U (42)where ε exp(2Πιn) +e set of univalent functions withn-fold symmetry (that is Ψn) has the expansion of the form

Ψn g(z) isin Ψ g(z) z + 1113944

infin

j1δnj+1z

nj+1 z isin U

⎧⎪⎨

⎪⎩(43)

Furthermore univalent function g(z) isin Ψn belongs toR(n)

pq if and only if

Dpq(g(z)) h(z) with h isin P(n) (44)

whereas


P(n)

h(z) isin P h(z) 1 + 1113944infin

j1dnjz

nj⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)

and function g(z) isin Ψn isin Slowast(n) ifRe(zgprime(z)g(z))gt 0 z isin U

Now if g isin Ψ(3) then g(z) z + δ4z4 + δ7z7 + middot middot middothenceHD41(g) δ24(δ

24 minus δ7) In the same way if g isin Ψ(2)

then g(z) z + δ3z3 + δ5z5 + middot middot middot clearly we can see thattwo-fold symmetric functions are odd SoHD41(g) δ3δ5δ7 minus δ33δ7 + δ23δ

25 minus δ35

Theorem 3 If g(z) is three-fold symmetric bounded turningfunction that is g(z) isinR(3)

pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le1

7[7]pq

(46)

Proof Firstly consider that g(z) isinR(3)pq then exist is a

function 1113957q isin Slowast(3) of the form z + α4z4 + α7z7 + middot middot middot suchthat (z1113957qprime(z)1113957q(z)) Dpqg(z) Since g(z) isinR(3)

pq and withthe use of (43) for n 3 it follows that

1 + 3α4z3

+ 6α7 minus 3α241113872 1113873z6

+ middot middot middot 1 +[4]pqδ4z3

+[7]pqδ7z6

+ middot middot middot

(47)

Identifying the coefficients we get

3α4 [4]pqδ4

6α7 minus 3α24 [7]pqδ7(48)

We already know 1113957q isin Slowast(3) then exist function q(z) of theform q(z) z + 1113936

infinj2 qjz

j isin Slowast such that 1113957q(z) [3]

radicq(z3)

+us

z + α4z4

+ α7z7

+ middot middot middot z +13q2z

4+

13q3 minus

19q221113874 1113875z

7+ middot middot middot

(49)

+erefore

α4 13q2

α7 13q3 minus

19q221113874 1113875

(50)

Now by rearranging (48) and (50) we get

δ4 q2

[4]pq

δ7 2q3 minus q

221113872 1113873

[7]pq

(51)

Since HD41(g) δ24(δ24 minus δ7) this implies

HD41(g)1113868111386811138681113868

1113868111386811138681113868le2

[4]2pq[7]pq

q22 q3 minus

[7]pq +[4]2pq1113872 1113873

2[4]2pq

q22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(52)

With the use of +eorem 1 in Section 2 where λ

([7]pq + [4]2pq)2[4]2pq isin [(58) (34)] for p 1 q⟶ 1minus we get our theorem proved

Theorem 4 If g is two-fold symmetric bounded turningfunction that is g isinR(2)

pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le8

[3]pq[5]pq[7]pq

(53)

Proof By the definition of two-fold symmetric function theHankel determinant can be written as

HD41(g) δ3δ5δ7 minus δ33δ7 + δ23δ25 minus δ35 (54)

Since g isinR(2)pq then exist is a function h isin P(2) such that

Dpq(g(z)) h(z) then the expansion of (43) and (45) forn 2 yields that

1 +[3]pqδ3z2

+[5]pqδ5z4

+[7]pqδ7z6

+ middot middot middot 1 + d2z2

+ r4z4

+ d6z6


there4δn 1

[n]pq

dnminus1

HD41(g)1113868111386811138681113868

1113868111386811138681113868 1

[3]pq[5]pq[7]pq

d2d4d6 minus1

[3]3pq[7]pq

d32d6 +

1[3]

2pq[5]

2pq

d22d

24 minus

1[5]

3pq

d34

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

le1

[3]pq[5]pq[7]pq

d2d6 minus[3]pq[7]pq

[5]2pq

d24

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868d4 minus

[5]pq

[3]2pq

d22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(55)


Now with the help of lemma in Section 2 we get ourdesired result as asserted by the statement

Remark 2 For p 1 q⟶ 1minus results of (29) (45) and (43)will coincide with results derived in [40]

Data Availability

+e required data are included in this article

Conflicts of Interest

+e authors declare that they have no conflicts of interest

References

[1] D O Jackson T Fukuda O Dunn and E Majors ldquoOnq-definite integralsrdquo Quarterly Journal of Pure and AppliedMathematics vol 14 1910

[2] F H Jackson ldquoq-difference equationsrdquo American Journal ofMathematics vol 32 no 4 pp 305ndash314 1910

[3] M E H Ismail E Merkes and D Styer ldquoA generalization ofstarlike functions Complex Variables +eory and Applica-tionrdquo An International Journal vol 14 no 1ndash4 pp 77ndash841990

[4] H M Srivastava and S Owa ldquoUnivalent functions fractionalcalculus and associated generalized hypergeometric func-tionsrdquo Fundamental -eory of Fractional Calculus vol 39pp 329ndash354 1989

[5] S Kanas and D Raducanu ldquoSome class of analytic functionsrelated to conic domainsrdquoMathematica Slovaca vol 64 no 5pp 1183ndash1196 2014

[6] M Arif H M Srivastava and S Umar ldquoSome applications ofa q-analogue of the Ruscheweyh type operator for multivalentfunctionsrdquo Revista de la Real Academia de Ciencias ExactasFısicas y Naturales Serie A Matematicas vol 113 no 2pp 1211ndash1221 2019b

[7] M Arif M U Haq and J L Liu ldquoA subfamily of univalentfunctions associated with q-analog of noor integral operatorrdquoJournal of Function Spaces vol 2018 Article ID 38189155 pages 2018a

[8] Q Khan M Arif M Raza G Srivastava H Tang andS u Rehman ldquoSome applications of a new integral operator inq-analog for multivalent functionsrdquo Mathematics vol 7no 12 p 1178 2019

[9] M Arif and B Ahmad ldquoNew subfamily of meromorphicmultivalent starlike functions in circular domain involvingq-differential operatorrdquo Mathematica Slovaca vol 68 no 5pp 1049ndash1056 2018

[10] H M Srivastava ldquoOperators of basic(or q-) calculas andfractional q-calculas and their applications in geometricfunction theory of complex analysisrdquo Iranian Journal ofScience and Technology Transactions A Science vol 44pp 1ndash18 2020

[11] M Arif O Barkub H Srivastava S Abdullah and S KhanldquoSome janowski type harmonic q-starlike functions associatedwith symmetrical pointsrdquo Mathematics vol 8 no 4 p 6292020

[12] L Shi Q Khan G Srivastava J-L Liu and M Arif ldquoA studyof multivalent q-starlike functions connected with circulardomainrdquo Mathematics vol 7 no 8 p 670 2019b

[13] R Chakrabarti and R Jagannathan ldquoA (p q)-oscillator re-alization of two-parameter quantum algebrasrdquo Journal of

Physics A Mathematical and General vol 24 no 13pp L711ndashL718 1991

[14] C Pommerenke ldquoOn the coefficients and hankel determi-nants of univalent functionsrdquo Journal of the London Math-ematical Society vol s1-41 no 1 pp 111ndash122 1966

[15] C Pommerenke ldquoOn the hankel determinants of univalentfunctionsrdquo Mathematika vol 14 no 1 pp 108ndash112 1967

[16] W K Hayman ldquoOn the second hankel determinant of meanunivalent functionsrdquo in Proceedings of the London Mathe-matical Society vol s3-18 no 1 pp 77ndash94 1968

[17] M Obradovic and N Tuneski ldquoHankel determinants ofsecond and third order for the class s of univalent functionsrdquo1912 httparxivorgabs191206439

[18] A Janteng S A Halim and M Darus ldquoCoefficient inequalityfor a function whose derivative has a positive real partrdquoJournal of Inequalties in Pureand Applied Mathematics vol 7no 2 pp 1ndash5 2006

[19] A Janteng S A Halim and M Darus ldquoHankel determinantfor starlike and convex functionsrdquo International Journal ofMathematical Analysis vol 1 no 13 pp 619ndash625 2007

[20] N E Cho B Kowalczyk O S Kwon A Lecko and Y J SimldquoSome coefficient inequalities related to the Hankel deter-minant for strongly starlike functions of order alphardquo Journalof Mathematical Inequalities vol 11 no 2 pp 429ndash439 2017

[21] N E Cho B Kowalczyk O S Kwon A Lecko and Y J Simldquo+e bounds of some determinants for starlike functions oforder alphardquo Bulletin of the Malaysian Mathematical SciencesSociety vol 41 no 1 pp 523ndash535 2018

[22] S K Lee V Ravichandran and S Supramaniam ldquoBounds forthe second hankel determinant of certain univalent func-tionsrdquo Journal of Inequalities and Applications vol 2013no 1 p 281 2013

[23] A Ebadian T Bulboaca N E Cho and E A AdeganildquoCoefficient bounds and differential subordinations for an-alytic functions associated with starlike functionsrdquo RACSAMvol 114 2020

[24] S Altınkaya and S Yalccedilın ldquoUpper bound of second hankeldeterminant for bi-bazilevic functionsrdquo MediterraneanJournal of Mathematics vol 13 no 6 pp 4081ndash4090 2016

[25] D Bansal ldquoUpper bound of second hankel determinant for anew class of analytic functionsrdquo Applied Mathematics Lettersvol 26 no 1 pp 103ndash107 2013

[26] M Ccedilaglar E Deniz and H M Srivastava ldquoSecond hankeldeterminant for certain subclasses of bi-univalent functionsrdquoTurkish Journal of Mathematics vol 41 no 3 pp 694ndash7062017

[27] S Kanas E A Adegani and A Zireh ldquoAn unified approach tosecond hankel determinant of bi-subordinate functionsrdquoMediterranean Journal of Mathematics vol 14 no 6 pp 1ndash122017

[28] M-S Liu J-F Xu and M Yang ldquoUpper bound of secondhankel determinant for certain subclasses of analytic func-tionsrdquo Abstract and Applied Analysis vol 2014 Article ID603180 10 pages 2014

[29] K O Babalola ldquoOn h_3 (1) hankel determinant for someclasses of univalent functionsrdquo Inequality -eory and Ap-plications vol 6 pp 1ndash7 2012

[30] S Altinkaya and S Yalccedilin ldquo+ird hankel determinant forbazilevic functionsrdquoAdvances inMath vol 5 pp 91ndash96 2016

[31] D Bansal S Maharana and J K Prajapat ldquo+ird orderhankel determinant for certain univalent functionsrdquo Journalof the Korean Mathematical Society vol 52 no 6pp 1139ndash1148 2015


[32] D V Krishna B Venkateswarlu and T RamReddy ldquo+irdhankel determinant for bounded turning functions of orderalphardquo Journal of the Nigerian Mathematical Society vol 34no 2 pp 121ndash127 2015

[33] M Raza and S N Malik ldquoUpper bound of the third hankeldeterminant for a class of analytic functions related withlemniscate of Bernoullirdquo Journal of Inequalities and Appli-cations vol 2013 no 1 pp 1ndash8 2013

[34] G Shanmugam B A Stephen and K O Babalola ldquo+irdhankel determinant for α-starlike functionsrdquo Gulf Journal ofMathematics vol 2 no 2 pp 107ndash113 2014

[35] P Zaprawa ldquo+ird hankel determinants for subclasses ofunivalent functionsrdquo Mediterranean Journal of Mathematicsvol 14 no 1 p 19 2017

[36] O S Kwon A Lecko and Y J Sim ldquo+e bound of the hankeldeterminant of the third kind for starlike functionsrdquo Bulletinof the Malaysian Mathematical Sciences Society vol 42 no 2pp 767ndash780 2019

[37] P Zaprawa M Obradovic and N Tuneski ldquo+ird hankeldeterminant for univalent starlike functions Revista de la RealAcademia de Ciencias Exactas Fısicas y Naturalesrdquo SerieA Matematicas vol 115 no 2 pp 1ndash6 2021

[38] B Kowalczyk A Lecko M Lecko and Y J Sim ldquo+e sharpbound of the third hankel determinant for some classes ofanalytic functionsrdquo Bulletin of the Korean MathematicalSociety vol 55 no 6 pp 1859ndash1868 2018

[39] A Lecko Y J Sim and B Smiarowska ldquo+e sharp bound ofthe hankel determinant of the third kind for starlike functionsof order 12rdquo Complex Analysis and Operator -eory vol 13no 5 pp 2231ndash2238 2019

[40] M Arif L Rani M Raza and P Zaprawa ldquoFourth hankeldeterminant for the family of functions with boundedturningrdquo Bulletin of the Korean Mathematical Society vol 55no 6 pp 1703ndash1711 2018b

[41] G Kaur and G Singh ldquo4+ hankel determinant for$Alpha$ bounded turning functionrdquo Advances in Mathe-matics Scientific Journal vol 9 no 12 pp 10563ndash10567 2020

[42] M Arif M Raza H Tang S Hussain and H Khan ldquoHankeldeterminant of order three for familiar subsets of analyticfunctions related with sine functionrdquo Open Mathematicsvol 17 no 1 pp 1615ndash1630 2019a

[43] M Shafiq H M Srivastava N Khan Q Z Ahmad M Darusand S Kiran ldquoAn upper bound of the third hankel deter-minant for a subclass of q-starlike functions associated withk-fibonacci numbersrdquo Symmetry vol 12 no 6 2020

[44] L Shi I Ali M Arif N E Cho S Hussain and H Khan ldquoAstudy of third hankel determinant problem for certain sub-families of analytic functions involving cardioid domainrdquoMathematics vol 7 no 5 p 418 2019a

[45] H M Srivastava S Altınkaya and S Yalccedilın ldquoHankel de-terminant for a subclass of bi-univalent functions defined byusing a symmetric q-derivative operatorrdquo Filomat vol 32no 2 pp 503ndash516 2018

[46] H M Srivastava B Khan N Khan and Q Z AhmadldquoCoeffcient inequalities for q-starlike functions associatedwith the janowski functionsrdquo Hokkaido Mathematical Jour-nal vol 48 pp 407ndash425 2019

[47] H M Srivastava G Kaur and G Singh ldquoEstimates of thefourth hankel determinant for a class of analytic functionswith bounded turnings involving cardioid domainsrdquo Journalof Nonlinear and Convex Analysis vol 22 no 3 pp 511ndash5262021

[48] C Pommerenke Univalent Functions Vandenhoeck andRuprecht Gottingen Germany 1975


foundation is given in [4] by Srivastava After this de-velopment many researchers introduced and studiedsome useful operators in q-analog with the applications ofconvolution concepts For example Kanas and Raducanu[5] established the q-differential operator and then ex-amined the behavior of this operator in function theory+is operator was generalized further for multivalentanalytic functions by Arif et al [6] Analogous to q-dif-ferential operator Arif et al and Khan et al contributedthe integral operators for analytic and multivalent func-tions in [7 8] respectively Similarly in the article [9] theauthors developed and analyzed the operator in q-analogfor meromorphic functions Also see the survey typearticle [10] on quantum calculus and their applicationsWith the use of these operators many researchers werecontributed some good articles in this direction in thefield of geometric function theory see [11 12] +esubstitution of q by qp in q-calculus has given us(p q)-calculus which is the extension of q-calculusChakrabarti and Jagannathan [13] considered(p q)-integer

For 0lt qltple 1 and zne 0 the (p q)-derivative operatorDpqg of the function g(z) isin A is defined as

Dpqg(z) g(pz) minus g(qz)

(p minus q)z (4)

From the use (1) and (4) the following formulae areeasily obtained

Dpq g1(z) + g2(z)( 1113857 Dpqg1(z) + Dpqg2(z) (5)

and for a constant c

Dpq(cg(z)) cDpqg(z) (6)

Here we also note that Dpqg(z) gprime(z) when we takep 1 and q⟶ 1minus

limq⟶1minus

Dpq(g(z)) limq⟶1minus

g(z) minus g(qz)

(1 minus q)z gprime(z) (7)

Further by using (1) and (4) we have

Dpq(g(z)) 1 + 1113944infin

λ2[λ]pqδλz

λminus 1 (8)

with

[λ]pq pλ

minus qλ

p minus q p

λminus 1 1 + 1113944λminus1

l1

q

p1113888 1113889

l

⎛⎝ ⎞⎠ (9)

Let us define the class Rpq which consists all thefunctions g isin A satisfying

Re Dpq(g(z))1113872 1113873gt 0 z isin U (10)

+e Hankel determinant HDnλ(g) withn λ isin N 1 2 and δ1 1 for a function g isin Ψ of theseries form (1) was given by Pommerenke [14 15] as

HDnλ(g)

δλ δλ+1 middot middot middot δλ+nminus1

δλ+1 δλ+2 middot middot middot δλ+n

⋮ ⋮ ⋮ ⋮

δλ+nminus1 δλ+n middot middot middot δλ+2nminus2

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(11)

In particular the following determinants are known asthe first- second- and third-order Hankel determinantsrespectively

HD21(g) 1 δ2δ2 δ3

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 δ3 minus δ22

HD22(g) δ2 δ3δ3 δ4

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 δ2δ4 minus δ23

HD31(g)

1 δ2 δ3δ2 δ3 δ4δ3 δ4 δ5

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

δ3 δ2δ4 minus δ231113872 1113873 minus δ4 δ4 minus δ2δ3( 1113857

+ δ5 δ3 minus δ221113872 1113873

(12)

+ere are comparatively few observations in literature inrelation to the Hankel determinant for the function g whichbelongs to the general family Ψ For the function g isin Ψ thebest established sharp inequality is |HD2λg|le μ

λ

radic where μ

is the absolute constant which is due to Hayman [16]Further for the same class Ψ it was obtained in [17] that

HD22(g)1113868111386811138681113868

1113868111386811138681113868le μ1 for 1le μ1 le113

HD31(g)1113868111386811138681113868

1113868111386811138681113868le μ2 for49le μ2 le

32 +285

radic

15

(13)

In a given family of functions the problem of calculatingthe bounds probably sharp of Hankel determinantsattracted the minds of several mathematicians For examplethe sharp bound of |HD22(g)| for the subfamilies K Slowastand R (family of bounded turning functions) of the set Ψwas calculated by Janteng et al [18 19] +ese estimates are

HD22(g)1113868111386811138681113868

1113868111386811138681113868le

18 forf isinK

1 forf isin Slowast

49 forf isinR

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(14)

For the following two families Slowast(β) (0le βlt 1) of star-like functions of order β and for SSlowast(β) (0lt βle 1) ofstrongly star-like functions of order β the authors computedin [20 21] that |HD22(g)| is bounded by (1 minus β)2 and β2respectively +e exact bound for the family of Ma-Mindastar-like functions was measured in [22] see also [23] Formore work on |HD22(g)| see references [24ndash28]

It is quite clear from the formulae given in (12) that thecalculation of |HD31(g)| is far more challenging compared



HD31(g)1113868111386811138681113868

1113868111386811138681113868le

49540

forg isinK

1 forg isin Slowast

4160

forg isinR

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


HD31(g)1113868111386811138681113868

1113868111386811138681113868le

4135

for g isinK


12

1113874 1113875

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)



HD41(g)1113868111386811138681113868

1113868111386811138681113868le 078050 (17)


2 A Set of Results



dλ1113868111386811138681113868

1113868111386811138681113868|2 for λ isin N (18)

dλ+k minus μdλdk

11138681113868111386811138681113868111386811138681113868le 2 for 0le μle 1 (19)

dpdq minus drds

11138681113868111386811138681113868

11138681113868111386811138681113868le 4 forp + q r + s (20)



δ22 δ3 minus μδ221113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868


12(2μ minus 1)

for μ isin5834

1113876 1113877

14(1 minus μ)

for μ isin3478

1113876 1113877


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)




(22)



+us

1 + 1113944infin

λ2[λ]pqδλz

λminus 1⎛⎝ ⎞⎠ 1 + 1113944infin

λ1dλz

λ⎛⎝ ⎞⎠ (24)


δλ dλminus1

[λ]pq

(25)


HD31(g) 1

[3]pq[5]pq

d2d4 minus1

[4]2pq

d23 +

2[2]pq[3]pq[4]pq

d1d2d3 minus1

[2]2pq[5]pq

d21d4 minus

1[3]

3pq

d32 (26)



HD31(g) 1

[2]2pq[5]pq

d4 d2 minus d211113872 1113873 minus

1[4]

2pq

d3 d3 minus d1d2( 1113857 +1

[3]3pq

d2 d4 minus d221113872 1113873

minus2

[2]pq[3]pq[4]pq

minus1

[4]2pq

⎛⎝ ⎞⎠d2 d4 minus d1d3( 1113857

+1

[3]pq[5]pq

minus1

[2]2pq[5]pq

minus1

[3]3pq

+2

[2]pq[3]pq[4]pq

minus1

[4]2pq

⎛⎝ ⎞⎠d2d4

(27)


|HD31(g)|le 44

[2]pq[3]pq[4]pq

+1

[3]pq[5]pq

minus1

[4]2pq

⎛⎝ ⎞⎠

(28)







(30)



(32)


HD41(g)1113868111386811138681113868

1113868111386811138681113868le 81

[3]pq[7]pq

σ1 +1

[4]2pq

σ2 +1

[5]2pq

σ3 +1

[6]pq

σ4⎛⎝ ⎞⎠

(33)

where

σ1 4

[2]pq[4]pq

+1

[5]pq

1113888 1113889

σ2 2

[2]pq[6]pq

minus1

[7]pq

+4

[3]pq[5]pq

1113888 1113889

σ3 2

[2]pq[4]pq

+2

[3]2pq

minus1

[5]pq

⎛⎝ ⎞⎠

σ4 2

[3]2pq[4]pq

+2

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠

(34)


1 + 1113944infin

λ2[λ]pqδλz

λminus 1⎛⎝ ⎞⎠ 1 + 1113944infin

λ1dλz

λ⎛⎝ ⎞⎠ (35)

+erefore



Λ1 1

[3]pq[6]pq

d2d5 minus1

[4]pq[5]pq

d3d4 minus1

[2]2pq[6]pq

d21d5 +

1[2]pq[3]pq[5]pq

d1d2d4 +1

[2]pq[4]2pq

d1d23

minus1

[3]2pq[4]pq

d22d3

Λ2 1

[4]pq[6]pq

d3d5 minus1

[5]2pq

d24 minus

1[2]pq[3]pq[6]pq

d1d2d5 +1

[2]pq[4]pq[5]pq

d1d3d4 +1

[3]2pq[5]pq

d22d4

minus1

[3]pq[4]2pq

d2d23

Λ3 1

[2]pq[4]pq[6]pq

d1d3d5 minus1

[2]pq[5]2pq

d1d24 minus

1[3]

2pq[6]pq

d22d5 +

2[3]pq[4]pq[5]pq

d2d3d4 minus1

[4]3pq

d33

(37)



Λ1 1

[2]2pq[6]pq

d5 d2 minus d211113872 1113873 +

1[3]

2pq[4]pq

d3 d4 minus d221113872 1113873 minus

1[2]pq[4]

2pq

d3 d4 minus d1d3( 1113857

minus1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

⎛⎝ ⎞⎠d4 d3 minus d1d2( 1113857

+1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

minus1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2 d5 minus d1d4( 1113857

+1

[6]pq[3]pq

minus1

[6]pq[2]2pq

minus1

[4]pq[5]pq

minus1

[3]2pq[4]pq

+1

[2]pq[4]2pq

+1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2d5

Λ2 1

[2]pq[3]pq[6]pq


[3]2pq[5]pq

d4 d4 minus d221113872 1113873 +

1[3]pq[4]

2pq

d3 d5 minus d2d3( 1113857

minus1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3 d5 minus d1d4( 1113857 minus1

[5]2pq

minus1

[3]2pq[5]pq

⎛⎝ ⎞⎠d4 d4 minus d1d3( 1113857

+1

[4]pq[6]pq

minus1

[2]pq[3]pq[6]pq

minus1

[3]pq[4]2pq

+1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3d5

Λ3 1

[3]2pq[6]pq

d5 d4 minus d221113872 1113873 minus

1[2]pq[4]pq[6]pq

d5 d4 minus d1d3( 1113857 +1

[4]3pq

d3 d6 minus d231113872 1113873

minus1

[4]3pq

d3 d6 minus d2d4( 1113857 +1

[2]pq[5]2pq


[3]pq[4]pq[5]pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4 d5 minus d2d3( 1113857

+1

[2]pq[4]pq[6]pq

+2

[3]pq[4]pq[5]pq

minus1

[3]2pq[6]pq

minus1

[2]pq[5]2pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4d5

(38)


Λ11113868111386811138681113868

1113868111386811138681113868le 42

[3]2pq[4]pq

+1

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠ (39)

Λ21113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[5]pq

+2

[3]2pq[5]pq

+1

[4]pq[6]pq

minus1

[5]2pq

⎛⎝ ⎞⎠

(40)

Λ31113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[6]pq

+4

[3]pq[4]pq[5]pq

1113888 1113889 (41)






Ψn g(z) isin Ψ g(z) z + 1113944

infin

j1δnj+1z

nj+1 z isin U

⎧⎪⎨

⎪⎩(43)


pq if and only if


whereas


P(n)


j1dnjz

nj⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)





25 minus δ35


pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le1

7[7]pq

(46)




1 + 3α4z3

+ 6α7 minus 3α241113872 1113873z6


+[7]pqδ7z6


(47)


3α4 [4]pqδ4

6α7 minus 3α24 [7]pqδ7(48)


infinj2 qjz


radicq(z3)

+us

z + α4z4

+ α7z7


4+

13q3 minus

19q221113874 1113875z


(49)

+erefore

α4 13q2

α7 13q3 minus

19q221113874 1113875

(50)


δ4 q2

[4]pq

δ7 2q3 minus q

221113872 1113873

[7]pq

(51)


HD41(g)1113868111386811138681113868

1113868111386811138681113868le2

[4]2pq[7]pq

q22 q3 minus

[7]pq +[4]2pq1113872 1113873

2[4]2pq

q22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(52)




pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le8

[3]pq[5]pq[7]pq

(53)





1 +[3]pqδ3z2

+[5]pqδ5z4

+[7]pqδ7z6


+ r4z4

+ d6z6


there4δn 1

[n]pq

dnminus1

HD41(g)1113868111386811138681113868

1113868111386811138681113868 1

[3]pq[5]pq[7]pq

d2d4d6 minus1

[3]3pq[7]pq

d32d6 +

1[3]

2pq[5]

2pq

d22d

24 minus

1[5]

3pq

d34

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

le1

[3]pq[5]pq[7]pq


[5]2pq

d24

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868d4 minus

[5]pq

[3]2pq

d22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(55)




Data Availability




References





















































HD31(g)1113868111386811138681113868

1113868111386811138681113868le

49540

forg isinK

1 forg isin Slowast

4160

forg isinR

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)


HD31(g)1113868111386811138681113868

1113868111386811138681113868le

4135

for g isinK


12

1113874 1113875

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(16)



HD41(g)1113868111386811138681113868

1113868111386811138681113868le 078050 (17)


2 A Set of Results



dλ1113868111386811138681113868

1113868111386811138681113868|2 for λ isin N (18)

dλ+k minus μdλdk

11138681113868111386811138681113868111386811138681113868le 2 for 0le μle 1 (19)

dpdq minus drds

11138681113868111386811138681113868

11138681113868111386811138681113868le 4 forp + q r + s (20)



δ22 δ3 minus μδ221113872 111387311138681113868111386811138681113868

11138681113868111386811138681113868


12(2μ minus 1)

for μ isin5834

1113876 1113877

14(1 minus μ)

for μ isin3478

1113876 1113877


⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)




(22)



+us

1 + 1113944infin

λ2[λ]pqδλz

λminus 1⎛⎝ ⎞⎠ 1 + 1113944infin

λ1dλz

λ⎛⎝ ⎞⎠ (24)


δλ dλminus1

[λ]pq

(25)


HD31(g) 1

[3]pq[5]pq

d2d4 minus1

[4]2pq

d23 +

2[2]pq[3]pq[4]pq

d1d2d3 minus1

[2]2pq[5]pq

d21d4 minus

1[3]

3pq

d32 (26)



HD31(g) 1

[2]2pq[5]pq

d4 d2 minus d211113872 1113873 minus

1[4]

2pq

d3 d3 minus d1d2( 1113857 +1

[3]3pq

d2 d4 minus d221113872 1113873

minus2

[2]pq[3]pq[4]pq

minus1

[4]2pq

⎛⎝ ⎞⎠d2 d4 minus d1d3( 1113857

+1

[3]pq[5]pq

minus1

[2]2pq[5]pq

minus1

[3]3pq

+2

[2]pq[3]pq[4]pq

minus1

[4]2pq

⎛⎝ ⎞⎠d2d4

(27)


|HD31(g)|le 44

[2]pq[3]pq[4]pq

+1

[3]pq[5]pq

minus1

[4]2pq

⎛⎝ ⎞⎠

(28)







(30)



(32)


HD41(g)1113868111386811138681113868

1113868111386811138681113868le 81

[3]pq[7]pq

σ1 +1

[4]2pq

σ2 +1

[5]2pq

σ3 +1

[6]pq

σ4⎛⎝ ⎞⎠

(33)

where

σ1 4

[2]pq[4]pq

+1

[5]pq

1113888 1113889

σ2 2

[2]pq[6]pq

minus1

[7]pq

+4

[3]pq[5]pq

1113888 1113889

σ3 2

[2]pq[4]pq

+2

[3]2pq

minus1

[5]pq

⎛⎝ ⎞⎠

σ4 2

[3]2pq[4]pq

+2

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠

(34)


1 + 1113944infin

λ2[λ]pqδλz

λminus 1⎛⎝ ⎞⎠ 1 + 1113944infin

λ1dλz

λ⎛⎝ ⎞⎠ (35)

+erefore



Λ1 1

[3]pq[6]pq

d2d5 minus1

[4]pq[5]pq

d3d4 minus1

[2]2pq[6]pq

d21d5 +

1[2]pq[3]pq[5]pq

d1d2d4 +1

[2]pq[4]2pq

d1d23

minus1

[3]2pq[4]pq

d22d3

Λ2 1

[4]pq[6]pq

d3d5 minus1

[5]2pq

d24 minus

1[2]pq[3]pq[6]pq

d1d2d5 +1

[2]pq[4]pq[5]pq

d1d3d4 +1

[3]2pq[5]pq

d22d4

minus1

[3]pq[4]2pq

d2d23

Λ3 1

[2]pq[4]pq[6]pq

d1d3d5 minus1

[2]pq[5]2pq

d1d24 minus

1[3]

2pq[6]pq

d22d5 +

2[3]pq[4]pq[5]pq

d2d3d4 minus1

[4]3pq

d33

(37)



Λ1 1

[2]2pq[6]pq

d5 d2 minus d211113872 1113873 +

1[3]

2pq[4]pq

d3 d4 minus d221113872 1113873 minus

1[2]pq[4]

2pq

d3 d4 minus d1d3( 1113857

minus1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

⎛⎝ ⎞⎠d4 d3 minus d1d2( 1113857

+1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

minus1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2 d5 minus d1d4( 1113857

+1

[6]pq[3]pq

minus1

[6]pq[2]2pq

minus1

[4]pq[5]pq

minus1

[3]2pq[4]pq

+1

[2]pq[4]2pq

+1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2d5

Λ2 1

[2]pq[3]pq[6]pq


[3]2pq[5]pq

d4 d4 minus d221113872 1113873 +

1[3]pq[4]

2pq

d3 d5 minus d2d3( 1113857

minus1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3 d5 minus d1d4( 1113857 minus1

[5]2pq

minus1

[3]2pq[5]pq

⎛⎝ ⎞⎠d4 d4 minus d1d3( 1113857

+1

[4]pq[6]pq

minus1

[2]pq[3]pq[6]pq

minus1

[3]pq[4]2pq

+1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3d5

Λ3 1

[3]2pq[6]pq

d5 d4 minus d221113872 1113873 minus

1[2]pq[4]pq[6]pq

d5 d4 minus d1d3( 1113857 +1

[4]3pq

d3 d6 minus d231113872 1113873

minus1

[4]3pq

d3 d6 minus d2d4( 1113857 +1

[2]pq[5]2pq


[3]pq[4]pq[5]pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4 d5 minus d2d3( 1113857

+1

[2]pq[4]pq[6]pq

+2

[3]pq[4]pq[5]pq

minus1

[3]2pq[6]pq

minus1

[2]pq[5]2pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4d5

(38)


Λ11113868111386811138681113868

1113868111386811138681113868le 42

[3]2pq[4]pq

+1

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠ (39)

Λ21113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[5]pq

+2

[3]2pq[5]pq

+1

[4]pq[6]pq

minus1

[5]2pq

⎛⎝ ⎞⎠

(40)

Λ31113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[6]pq

+4

[3]pq[4]pq[5]pq

1113888 1113889 (41)






Ψn g(z) isin Ψ g(z) z + 1113944

infin

j1δnj+1z

nj+1 z isin U

⎧⎪⎨

⎪⎩(43)


pq if and only if


whereas


P(n)


j1dnjz

nj⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)





25 minus δ35


pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le1

7[7]pq

(46)




1 + 3α4z3

+ 6α7 minus 3α241113872 1113873z6


+[7]pqδ7z6


(47)


3α4 [4]pqδ4

6α7 minus 3α24 [7]pqδ7(48)


infinj2 qjz


radicq(z3)

+us

z + α4z4

+ α7z7


4+

13q3 minus

19q221113874 1113875z


(49)

+erefore

α4 13q2

α7 13q3 minus

19q221113874 1113875

(50)


δ4 q2

[4]pq

δ7 2q3 minus q

221113872 1113873

[7]pq

(51)


HD41(g)1113868111386811138681113868

1113868111386811138681113868le2

[4]2pq[7]pq

q22 q3 minus

[7]pq +[4]2pq1113872 1113873

2[4]2pq

q22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(52)




pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le8

[3]pq[5]pq[7]pq

(53)





1 +[3]pqδ3z2

+[5]pqδ5z4

+[7]pqδ7z6


+ r4z4

+ d6z6


there4δn 1

[n]pq

dnminus1

HD41(g)1113868111386811138681113868

1113868111386811138681113868 1

[3]pq[5]pq[7]pq

d2d4d6 minus1

[3]3pq[7]pq

d32d6 +

1[3]

2pq[5]

2pq

d22d

24 minus

1[5]

3pq

d34

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

le1

[3]pq[5]pq[7]pq


[5]2pq

d24

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868d4 minus

[5]pq

[3]2pq

d22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(55)




Data Availability




References





















































HD31(g) 1

[2]2pq[5]pq

d4 d2 minus d211113872 1113873 minus

1[4]

2pq

d3 d3 minus d1d2( 1113857 +1

[3]3pq

d2 d4 minus d221113872 1113873

minus2

[2]pq[3]pq[4]pq

minus1

[4]2pq

⎛⎝ ⎞⎠d2 d4 minus d1d3( 1113857

+1

[3]pq[5]pq

minus1

[2]2pq[5]pq

minus1

[3]3pq

+2

[2]pq[3]pq[4]pq

minus1

[4]2pq

⎛⎝ ⎞⎠d2d4

(27)


|HD31(g)|le 44

[2]pq[3]pq[4]pq

+1

[3]pq[5]pq

minus1

[4]2pq

⎛⎝ ⎞⎠

(28)







(30)



(32)


HD41(g)1113868111386811138681113868

1113868111386811138681113868le 81

[3]pq[7]pq

σ1 +1

[4]2pq

σ2 +1

[5]2pq

σ3 +1

[6]pq

σ4⎛⎝ ⎞⎠

(33)

where

σ1 4

[2]pq[4]pq

+1

[5]pq

1113888 1113889

σ2 2

[2]pq[6]pq

minus1

[7]pq

+4

[3]pq[5]pq

1113888 1113889

σ3 2

[2]pq[4]pq

+2

[3]2pq

minus1

[5]pq

⎛⎝ ⎞⎠

σ4 2

[3]2pq[4]pq

+2

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠

(34)


1 + 1113944infin

λ2[λ]pqδλz

λminus 1⎛⎝ ⎞⎠ 1 + 1113944infin

λ1dλz

λ⎛⎝ ⎞⎠ (35)

+erefore



Λ1 1

[3]pq[6]pq

d2d5 minus1

[4]pq[5]pq

d3d4 minus1

[2]2pq[6]pq

d21d5 +

1[2]pq[3]pq[5]pq

d1d2d4 +1

[2]pq[4]2pq

d1d23

minus1

[3]2pq[4]pq

d22d3

Λ2 1

[4]pq[6]pq

d3d5 minus1

[5]2pq

d24 minus

1[2]pq[3]pq[6]pq

d1d2d5 +1

[2]pq[4]pq[5]pq

d1d3d4 +1

[3]2pq[5]pq

d22d4

minus1

[3]pq[4]2pq

d2d23

Λ3 1

[2]pq[4]pq[6]pq

d1d3d5 minus1

[2]pq[5]2pq

d1d24 minus

1[3]

2pq[6]pq

d22d5 +

2[3]pq[4]pq[5]pq

d2d3d4 minus1

[4]3pq

d33

(37)



Λ1 1

[2]2pq[6]pq

d5 d2 minus d211113872 1113873 +

1[3]

2pq[4]pq

d3 d4 minus d221113872 1113873 minus

1[2]pq[4]

2pq

d3 d4 minus d1d3( 1113857

minus1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

⎛⎝ ⎞⎠d4 d3 minus d1d2( 1113857

+1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

minus1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2 d5 minus d1d4( 1113857

+1

[6]pq[3]pq

minus1

[6]pq[2]2pq

minus1

[4]pq[5]pq

minus1

[3]2pq[4]pq

+1

[2]pq[4]2pq

+1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2d5

Λ2 1

[2]pq[3]pq[6]pq


[3]2pq[5]pq

d4 d4 minus d221113872 1113873 +

1[3]pq[4]

2pq

d3 d5 minus d2d3( 1113857

minus1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3 d5 minus d1d4( 1113857 minus1

[5]2pq

minus1

[3]2pq[5]pq

⎛⎝ ⎞⎠d4 d4 minus d1d3( 1113857

+1

[4]pq[6]pq

minus1

[2]pq[3]pq[6]pq

minus1

[3]pq[4]2pq

+1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3d5

Λ3 1

[3]2pq[6]pq

d5 d4 minus d221113872 1113873 minus

1[2]pq[4]pq[6]pq

d5 d4 minus d1d3( 1113857 +1

[4]3pq

d3 d6 minus d231113872 1113873

minus1

[4]3pq

d3 d6 minus d2d4( 1113857 +1

[2]pq[5]2pq


[3]pq[4]pq[5]pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4 d5 minus d2d3( 1113857

+1

[2]pq[4]pq[6]pq

+2

[3]pq[4]pq[5]pq

minus1

[3]2pq[6]pq

minus1

[2]pq[5]2pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4d5

(38)


Λ11113868111386811138681113868

1113868111386811138681113868le 42

[3]2pq[4]pq

+1

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠ (39)

Λ21113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[5]pq

+2

[3]2pq[5]pq

+1

[4]pq[6]pq

minus1

[5]2pq

⎛⎝ ⎞⎠

(40)

Λ31113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[6]pq

+4

[3]pq[4]pq[5]pq

1113888 1113889 (41)






Ψn g(z) isin Ψ g(z) z + 1113944

infin

j1δnj+1z

nj+1 z isin U

⎧⎪⎨

⎪⎩(43)


pq if and only if


whereas


P(n)


j1dnjz

nj⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)





25 minus δ35


pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le1

7[7]pq

(46)




1 + 3α4z3

+ 6α7 minus 3α241113872 1113873z6


+[7]pqδ7z6


(47)


3α4 [4]pqδ4

6α7 minus 3α24 [7]pqδ7(48)


infinj2 qjz


radicq(z3)

+us

z + α4z4

+ α7z7


4+

13q3 minus

19q221113874 1113875z


(49)

+erefore

α4 13q2

α7 13q3 minus

19q221113874 1113875

(50)


δ4 q2

[4]pq

δ7 2q3 minus q

221113872 1113873

[7]pq

(51)


HD41(g)1113868111386811138681113868

1113868111386811138681113868le2

[4]2pq[7]pq

q22 q3 minus

[7]pq +[4]2pq1113872 1113873

2[4]2pq

q22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(52)




pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le8

[3]pq[5]pq[7]pq

(53)





1 +[3]pqδ3z2

+[5]pqδ5z4

+[7]pqδ7z6


+ r4z4

+ d6z6


there4δn 1

[n]pq

dnminus1

HD41(g)1113868111386811138681113868

1113868111386811138681113868 1

[3]pq[5]pq[7]pq

d2d4d6 minus1

[3]3pq[7]pq

d32d6 +

1[3]

2pq[5]

2pq

d22d

24 minus

1[5]

3pq

d34

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

le1

[3]pq[5]pq[7]pq


[5]2pq

d24

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868d4 minus

[5]pq

[3]2pq

d22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(55)




Data Availability




References





















































Λ1 1

[2]2pq[6]pq

d5 d2 minus d211113872 1113873 +

1[3]

2pq[4]pq

d3 d4 minus d221113872 1113873 minus

1[2]pq[4]

2pq

d3 d4 minus d1d3( 1113857

minus1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

⎛⎝ ⎞⎠d4 d3 minus d1d2( 1113857

+1

[4]pq[5]pq

+1

[3]2pq[4]pq

minus1

[2]pq[4]2pq

minus1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2 d5 minus d1d4( 1113857

+1

[6]pq[3]pq

minus1

[6]pq[2]2pq

minus1

[4]pq[5]pq

minus1

[3]2pq[4]pq

+1

[2]pq[4]2pq

+1

[2]pq[3]pq[5]pq

⎛⎝ ⎞⎠d2d5

Λ2 1

[2]pq[3]pq[6]pq


[3]2pq[5]pq

d4 d4 minus d221113872 1113873 +

1[3]pq[4]

2pq

d3 d5 minus d2d3( 1113857

minus1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3 d5 minus d1d4( 1113857 minus1

[5]2pq

minus1

[3]2pq[5]pq

⎛⎝ ⎞⎠d4 d4 minus d1d3( 1113857

+1

[4]pq[6]pq

minus1

[2]pq[3]pq[6]pq

minus1

[3]pq[4]2pq

+1

[2]pq[4]pq[5]pq

minus1

[5]2pq

+1

[3]2pq[5]pq

⎛⎝ ⎞⎠d3d5

Λ3 1

[3]2pq[6]pq

d5 d4 minus d221113872 1113873 minus

1[2]pq[4]pq[6]pq

d5 d4 minus d1d3( 1113857 +1

[4]3pq

d3 d6 minus d231113872 1113873

minus1

[4]3pq

d3 d6 minus d2d4( 1113857 +1

[2]pq[5]2pq


[3]pq[4]pq[5]pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4 d5 minus d2d3( 1113857

+1

[2]pq[4]pq[6]pq

+2

[3]pq[4]pq[5]pq

minus1

[3]2pq[6]pq

minus1

[2]pq[5]2pq

minus1

[4]3pq

⎛⎝ ⎞⎠d4d5

(38)


Λ11113868111386811138681113868

1113868111386811138681113868le 42

[3]2pq[4]pq

+1

[4]pq[5]pq

+1

[3]pq[6]pq

⎛⎝ ⎞⎠ (39)

Λ21113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[5]pq

+2

[3]2pq[5]pq

+1

[4]pq[6]pq

minus1

[5]2pq

⎛⎝ ⎞⎠

(40)

Λ31113868111386811138681113868

1113868111386811138681113868le 42

[2]pq[4]pq[6]pq

+4

[3]pq[4]pq[5]pq

1113888 1113889 (41)






Ψn g(z) isin Ψ g(z) z + 1113944

infin

j1δnj+1z

nj+1 z isin U

⎧⎪⎨

⎪⎩(43)


pq if and only if


whereas


P(n)


j1dnjz

nj⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)





25 minus δ35


pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le1

7[7]pq

(46)




1 + 3α4z3

+ 6α7 minus 3α241113872 1113873z6


+[7]pqδ7z6


(47)


3α4 [4]pqδ4

6α7 minus 3α24 [7]pqδ7(48)


infinj2 qjz


radicq(z3)

+us

z + α4z4

+ α7z7


4+

13q3 minus

19q221113874 1113875z


(49)

+erefore

α4 13q2

α7 13q3 minus

19q221113874 1113875

(50)


δ4 q2

[4]pq

δ7 2q3 minus q

221113872 1113873

[7]pq

(51)


HD41(g)1113868111386811138681113868

1113868111386811138681113868le2

[4]2pq[7]pq

q22 q3 minus

[7]pq +[4]2pq1113872 1113873

2[4]2pq

q22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(52)




pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le8

[3]pq[5]pq[7]pq

(53)





1 +[3]pqδ3z2

+[5]pqδ5z4

+[7]pqδ7z6


+ r4z4

+ d6z6


there4δn 1

[n]pq

dnminus1

HD41(g)1113868111386811138681113868

1113868111386811138681113868 1

[3]pq[5]pq[7]pq

d2d4d6 minus1

[3]3pq[7]pq

d32d6 +

1[3]

2pq[5]

2pq

d22d

24 minus

1[5]

3pq

d34

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

le1

[3]pq[5]pq[7]pq


[5]2pq

d24

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868d4 minus

[5]pq

[3]2pq

d22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(55)




Data Availability




References




















































P(n)


j1dnjz

nj⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (45)





25 minus δ35


pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le1

7[7]pq

(46)




1 + 3α4z3

+ 6α7 minus 3α241113872 1113873z6


+[7]pqδ7z6


(47)


3α4 [4]pqδ4

6α7 minus 3α24 [7]pqδ7(48)


infinj2 qjz


radicq(z3)

+us

z + α4z4

+ α7z7


4+

13q3 minus

19q221113874 1113875z


(49)

+erefore

α4 13q2

α7 13q3 minus

19q221113874 1113875

(50)


δ4 q2

[4]pq

δ7 2q3 minus q

221113872 1113873

[7]pq

(51)


HD41(g)1113868111386811138681113868

1113868111386811138681113868le2

[4]2pq[7]pq

q22 q3 minus

[7]pq +[4]2pq1113872 1113873

2[4]2pq

q22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(52)




pq then

HD41(g)1113868111386811138681113868

1113868111386811138681113868le8

[3]pq[5]pq[7]pq

(53)





1 +[3]pqδ3z2

+[5]pqδ5z4

+[7]pqδ7z6


+ r4z4

+ d6z6


there4δn 1

[n]pq

dnminus1

HD41(g)1113868111386811138681113868

1113868111386811138681113868 1

[3]pq[5]pq[7]pq

d2d4d6 minus1

[3]3pq[7]pq

d32d6 +

1[3]

2pq[5]

2pq

d22d

24 minus

1[5]

3pq

d34

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

le1

[3]pq[5]pq[7]pq


[5]2pq

d24

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868d4 minus

[5]pq

[3]2pq

d22

⎛⎝ ⎞⎠

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868

(55)




Data Availability




References






















































Data Availability




References






































































Documents

A Study of Third and Fourth Hankel Determinant Problem for