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Research ArticleSharp Bounds for Toader Mean in terms of Arithmetic andSecond Contraharmonic Means
Wei-Mao Qian1 Ying-Qing Song2 Xiao-Hui Zhang2 and Yu-Ming Chu2
1School of Distance Education Huzhou Broadcast and TV University Huzhou 313000 China2School of Mathematics and Computation Science Hunan City University Yiyang 413000 China
Correspondence should be addressed to Yu-Ming Chu chuyuming2005126com
Received 25 April 2015 Accepted 10 September 2015
Academic Editor Lars E Persson
Copyright copy 2015 Wei-Mao Qian et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We present the best possible parameters 1205821 1205831isin R and 120582
2 1205832isin (12 1) such that double inequalities 120582
1119862(119886 119887) + (1 minus 120582
1)119860(119886 119887) lt
119879(119886 119887) lt 1205831119862(119886 119887) + (1 minus 120583
1)119860(119886 119887) 119862[120582
2119886 + (1 minus 120582
2)119887 1205822119887 + (1 minus 120582
2)119886] lt 119879(119886 119887) lt 119862[120583
2119886 + (1 minus 120583
2)119887 1205832119887 + (1 minus 120583
2)119886] hold for
all 119886 119887 gt 0 with 119886 = 119887 where 119860(119886 119887) = (119886 + 119887)2 119862(119886 119887) = (1198863+ 1198873)(1198862+ 1198872) and 119879(119886 119887) = 2 int
1205872
0
radic1198862cos2120579 + 1198872sin2120579119889120579120587 are thearithmetic second contraharmonic and Toader means of 119886 and 119887 respectively
1 Introduction
For 119886 119887 gt 0 the Toader mean 119879(119886 119887) [1] second contrahar-monic mean 119862(119886 119887) and arithmetic mean 119860(119886 119887) of 119886 and 119887
are given by
119879 (119886 119887) =2
120587int
1205872
0
radic1198862cos2120579 + 1198872sin2120579 119889120579
=
2119886E(radic1 minus (119887119886)2)
120587 119886 gt 119887
2119887E(radic1 minus (119886119887)2)
120587 119886 lt 119887
119886 119886 = 119887
(1)
119862 (119886 119887) =1198863+ 1198873
1198862 + 1198872
119860 (119886 119887) =119886 + 119887
2
(2)
respectively where E(119903) = int1205872
0(1 minus 119903
2sin2119905)12119889119905 (119903 isin (0 1))
is the complete elliptic integral of the second kindTheToader
mean 119879(119886 119887) is well known in mathematical literature formany years it satisfies
119879 (119886 119887) = 119877119864(1198862 1198872)
119879 (1 119903) =2
120587E (radic1 minus 1199032)
(3)
for all 119886 119887 gt 0 and 0 lt 119903 lt 1 where
119877119864(119886 119887) =
1
120587int
infin
0
[119886 (119905 + 119887) + 119887 (119905 + 119886)] 119905
(119905 + 119886)32
(119905 + 119887)32
119889119905 (4)
stands for the symmetric complete elliptic integral of thesecond kind (see [2ndash4]) therefore it cannot be expressed interms of the elementary transcendental functions
Recently the Toader mean 119879(119886 119887) has been the subject ofintensive research In particular many remarkable inequali-ties for the Toader mean can be found in the literature [5ndash9]
Let 119901 isin R 119902 isin [0 1] and 119886 119887 gt 0 Then the 119901thpower mean 119872
119901(119886 119887) 119901th Gini mean 119866
119901(119886 119887) 119901th Lehmer
Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 452823 5 pageshttpdxdoiorg1011552015452823
2 Journal of Function Spaces
mean 119871119901(119886 119887) and 119902th generalized Seiffert mean 119878
119902(119886 119887) are
defined by
119872119901(119886 119887) =
(119886119901+ 119887119901
2)
1119901
119901 = 0
radic119886119887 119901 = 0
119866119901(119886 119887) =
(119886119901minus1
+ 119887119901minus1
119886 + 119887)
1(119901minus2)
119901 = 2
(119886119886119887119887)1(119886+119887)
119901 = 2
119871119901(119886 119887) =
119886119901+1
+ 119887119901+1
119886119901 + 119887119901
119878119902(119886 119887)
=
119901 (119886 minus 119887)
arctan [2119901 (119886 minus 119887) (119886 + 119887)] 0 lt 119901 le 1 119886 = 119887
119886 + 119887
2 119901 = 0 119886 = 119887
119886 119886 = 119887
(5)
respectively It is well known that119872119901(119886 119887)119866
119901(119886 119887) 119871
119901(119886 119887)
and 119878119902(119886 119887) are continuous and strictly increasing with
respect to 119901 isin R and 119902 isin [0 1] for fixed 119886 119887 gt 0 with 119886 = 119887respectively
Vuorinen [10] conjectured that inequality
11987232
(119886 119887) lt 119879 (119886 119887) (6)
holds for all 119886 119887 gt 0 with 119886 = 119887 This conjecture was provedby Qiu and Shen [11] and Barnard et al [12] respectively
Alzer and Qiu [13] presented a best possible upper powermean bound for the Toader mean as follows
119879 (119886 119887) lt 119872log 2(log120587minuslog 2) (119886 119887) (7)
for all 119886 119887 gt 0 with 119886 = 119887In [14 15] the authors found the best possible parameters
120572 120573 isin [0 1] and 120582 120583 isin R such that double inequalities119878120572(119886 119887) lt 119879(119886 119887) lt 119878
120573(119886 119887) and 119866
120582(119886 119887) lt 119879(119886 119887) lt
119866120583(119886 119887) hold for all 119886 119887 gt 0 with 119886 = 119887Chu and Wang [16] proved that double inequality
119871119901(119886 119887) lt 119879 (119886 119887) lt 119871
119902(119886 119887) (8)
holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 119901 le 0 and119902 ge 14
Inequality (8) leads to
119860 (119886 119887) = 1198710(119886 119887) lt 119879 (119886 119887) lt 119871
14(119886 119887) lt 119871
2(119886 119887)
= 119862 (119886 119887)
(9)
for all 119886 119887 gt 0 with 119886 = 119887Let 119886 119887 gt 0 with 119886 = 119887 be fixed and 119891(119909) = 119862[119909119886 + (1 minus
119909)119887 119909119887 + (1 minus 119909)119886] Then it is not difficult to verify that 119891(119909)is continuous and strictly increasing on [12 1] Note that
119891(1
2) = 119860 (119886 119887) lt 119879 (119886 119887) lt 119862 (119886 119887) = 119891 (1) (10)
Motivated by inequalities (9) and (10) it is natural to askwhat are the best possible parameters 120582
1 1205831isin R and 120582
2 1205832isin
(12 1) such that double inequalities
1205821119862 (119886 119887) + (1 minus 120582
1) 119860 (119886 119887) lt 119879 (119886 119887)
lt 1205831119862 (119886 119887) + (1 minus 120583
1) 119860 (119886 119887)
119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] lt 119879 (119886 119887)
lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886]
(11)
hold for all 119886 119887 gt 0 with 119886 = 119887 The main purpose of thispaper is to answer this question
2 Main Results
In order to prove our main results we need some basicknowledge and two lemmas which we present in this section
For 119903 isin (0 1) the complete elliptic integral K(119903) of thefirst kind is defined by
K (119903) = int
1205872
0
(1 minus 1199032sin2119905)
minus12
119889119905 (12)
We clearly see that
K (0+) = E (0
+) =
120587
2
E (1minus) = 1
(13)
and K(119903) and E(119903) satisfy formulas (see [17 Appendix E p474-475])
119889K (119903)
119889119903=E (119903) minus (1 minus 119903
2)K (119903)
119903 (1 minus 1199032)
119889E (119903)
119889119903=E (119903) minusK (119903)
119903
E(2radic119903
1 + 119903) =
2E (119903) minus (1 minus 1199032)K (119903)
1 + 119903
(14)
Lemma 1 (see [17 Theorem 125]) Let minusinfin lt 119886 lt 119887 lt
infin 119891 119892 [119886 119887] rarr (minusinfininfin) be continuous on [119886 119887] anddifferentiable on (119886 119887) and 119892
1015840(119909) = 0 on (119886 119887) If 1198911015840(119909)1198921015840(119909)
is increasing (decreasing) on (119886 119887) then so are
119891 (119909) minus 119891 (119886)
119892 (119909) minus 119892 (119886)
119891 (119909) minus 119891 (119887)
119892 (119909) minus 119892 (119887)
(15)
If 1198911015840(119909)1198921015840(119909) is strictly monotone then the monotonicity inthe conclusion is also strict
Lemma 2 (see [17 Theorem 321]) (1) Function 119903 997891rarr [E(119903) minus
(1 minus 1199032)K(119903)]119903
2 is strictly increasing from (0 1) to (1205874 1)(2) Function 119903 997891rarr (1 minus 119903
2)120582K(119903) is strictly decreasing from
(0 1) to (0 1205872) if 120582 ge 14
Journal of Function Spaces 3
Theorem 3 Double inequality
1205821119862 (119886 119887) + (1 minus 120582
1) 119860 (119886 119887) lt 119879 (119886 119887)
lt 1205831119862 (119886 119887) + (1 minus 120583
1) 119860 (119886 119887)
(16)
holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 1205821le 18 and
1205831ge 4120587 minus 1 = 02732
Proof Since 119860(119886 119887) 119879(119886 119887) and 119862(119886 119887) are symmetric andhomogeneous of degree 1 without loss of generality weassume that 119886 gt 119887 gt 0 Let 119903 = (119886 minus 119887)(119886 + 119887) isin (0 1)Then (1) and (2) lead to
119879 (119886 119887) =2119886
120587E(
2radic119903
1 + 119903)
=2119860 (119886 119887)
120587[2E (119903) minus (1 minus 119903
2)K (119903)]
(17)
119862 (119886 119887) = 119860 (119886 119887)1 + 3119903
2
1 + 1199032 (18)
We clearly see that inequality (16) is equivalent to
1205821lt
119879 (119886 119887) minus 119860 (119886 119887)
119862 (119886 119887) minus 119860 (119886 119887)lt 1205831 (19)
It follows from (17) and (18) that
119879 (119886 119887) minus 119860 (119886 119887)
119862 (119886 119887) minus 119860 (119886 119887)
=1
120587
2E (119903) minus (1 minus 1199032)K (119903) minus 1205872
1199032 (1 + 1199032)
(20)
Let
119891 (119903) =2E (119903) minus (1 minus 119903
2)K (119903) minus 1205872
1199032 (1 + 1199032)
1198911(119903) = 2E (119903) minus (1 minus 119903
2)K (119903) minus
120587
2
1198912(119903) =
1199032
1 + 1199032
(21)
Then simple computations lead to
1198911(0+) = 1198912(0+) = 0
1198911015840
1(119903)
11989110158402(119903)
=(1 + 119903
2)2
2
E (119903) minus (1 minus 1199032)K (119903)
1199032
(22)
From Lemmas 1 and 2 together with (21) and (22) weknow that 119891(119903) is strictly increasing on (0 1) and
119891 (0+) =
1198911015840
1(0+)
11989110158402(0+)
=120587
8
119891 (1minus) = 4 minus 120587
(23)
Therefore Theorem 3 follows from (19)ndash(21) and (23)together with the monotonicity of 119891(119903)
Theorem 4 Let 1205822 1205832isin (12 1) Then double inequality
119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] lt 119879 (119886 119887)
lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886]
(24)
holds for all 119886 119887 gt 0with 119886 = 119887 if and only if 1205822le 12+radic28 =
06767 and 1205832ge 12 + radic(4 minus 120587)(3120587 minus 4)2 = 06988
Proof Let 120582lowast
2= 12 + radic28 and 120583
lowast
2= 12 +
radic(4 minus 120587)(3120587 minus 4)2 We first prove that
119879 (119886 119887) gt 119862 [120582lowast
2119886 + (1 minus 120582
lowast
2) 119887 120582lowast
2119887 + (1 minus 120582
lowast
2) 119886] (25)
119879 (119886 119887) lt 119862 [120583lowast
2119886 + (1 minus 120583
lowast
2) 119887 120583lowast
2119887 + (1 minus 120583
lowast
2) 119886] (26)
for all 119886 119887 gt 0 with 119886 = 119887Without loss of generality we assume that 119886 gt 119887 Let 119903 =
(119886 minus 119887)(119886 + 119887) isin (0 1) and 119901 isin (12 1) Then (2) leads to
119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]
= 119860 (119886 119887)3 (2119901 minus 1)
2
1199032+ 1
(2119901 minus 1)2
1199032 + 1
(27)
It follows from (17) and (27) that
119879 (119886 119887) minus 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]
= 119860 (119886 119887) [2
120587(2E (119903) minus (1 minus 119903
2)K (119903))
+2
(2119901 minus 1)2
1199032 + 1
minus 3]
(28)
Let
119892 (119903) =2
120587(2E (119903) minus (1 minus 119903
2)K (119903))
+2
(2119901 minus 1)2
1199032 + 1
minus 3
(29)
1198921(119903) =
1
1199031198921015840(119903) (30)
4 Journal of Function Spaces
Then making use of Lemma 2 and simple computations leadto
119892 (0) = 0 (31)
119892 (1) =4
120587+
2
(2119901 minus 1)2
+ 1
minus 3 (32)
1198921(119903) =
2
120587
E (119903) minus (1 minus 1199032)K (119903)
1199032
minus4 (2119901 minus 1)
2
[(2119901 minus 1)2
1199032 + 1]2
(33)
1198921(0) =
1
2minus 4 (2119901 minus 1)
2
(34)
1198921(1) =
2
120587minus
4 (2119901 minus 1)2
[(2119901 minus 1)2
+ 1]2 (35)
We divide the proof into two cases
Case 1 Consider 119901 = 120582lowast
2= 12 + radic28 Then (34) becomes
1198921(0) = 0 (36)
It follows from Lemma 2(1) and (33) together with (36) that
1198921(119903) gt 0 (37)
for all 119903 isin (0 1)Therefore inequality (25) follows easily from (28)ndash(31)
and (37)
Case 2 Consider 119901 = 120583lowast
2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then
(32) (34) and (35) lead to
119892 (1) = 0 (38)
1198921(0) = minus
36 minus 11120587
6120587 minus 8lt 0 (39)
1198921(1) =
31205872minus 14120587 + 16
1205872gt 0 (40)
It follows from Lemma 2(1) (33) (39) and (40) that thereexists 119903
0isin (0 1) such that 119892
1(119903) lt 0 for 119903 isin (0 119903
0) and
1198921(119903) gt 0 for 119903 isin (119903
0 1)Then (30) leads to the conclusion that
119892(119903) is strictly decreasing on (0 1199030] and strictly increasing on
(1199030 1]Therefore inequality (26) follows easily from (28) (29)
(31) (38) and the piecewise monotonicity of 119892(119903)Next we prove that 120582
2= 120582lowast
2= 12 + radic28 is the best
possible parameter on (12 1) such that inequality
119879 (119886 119887) gt 119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] (41)
holds for all 119886 119887 gt 0 with 119886 = 119887Indeed if 120582lowast
2= 12 + radic28 lt 119901 lt 1 then (34) leads to
1198921(0) lt 0 and there exists 120575
1isin (0 1) such that
1198921(119903) lt 0 (42)
for 119903 isin (0 1205751)
Equations (28)ndash(31) and inequality (42) imply that
119879 (119886 119887) lt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (43)
for (119886 minus 119887)(119886 + 119887) isin (0 1205751)
Finally we prove that 1205832= 120583lowast
2= 12+radic(4 minus 120587)(3120587 minus 4)
2 is the best possible parameter on (12 1) such that doubleinequality
119879 (119886 119887) lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886] (44)
for all 119886 119887 gt 0 with 119886 = 119887In fact if 12 lt 119901 lt 120583
lowast
2= 12+radic(4 minus 120587)(3120587 minus 4)2 then
(32) leads to 119892(1) gt 0 and there exists 1205752isin (0 1) such that
119892 (119903) gt 0 (45)
for 119903 isin (1 minus 1205752 1)
Equations (28) and (29) together with inequality (45)imply that
119879 (119886 119887) gt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (46)
for (119886 minus 119887)(119886 + 119887) isin (1 minus 1205752 1)
Let 119903 isin (0 1) 119886 = 1 119887 = radic1 minus 1199032 1205821= 18 120583
1= 4120587 minus 1
1205822= 12 + radic28 and 120583
2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then
Theorems 3 and 4 lead to Corollary 5 as follows
Corollary 5 Double inequalities
120587 [1 + (1 minus 1199032)32
]
16 (2 minus 1199032)+
7120587 [1 + (1 minus 1199032)12
]
32lt E (119903)
lt
(4 minus 120587) [1 + (1 minus 1199032)32
]
2 (2 minus 1199032)+ (
120587
2minus 1) [1 + (1 minus 119903
2)12
]
120587
2
32 minus 211199032+ 21 (1 minus 119903
2)12
+ 11 (1 minus 1199032)32
36 minus 181199032 + 28 (1 minus 1199032)12
lt E (119903) lt120587
2
sdot3120587 minus 4 minus 3 (120587 minus 2) 119903
2+ 3 (120587 minus 2) (1 minus 119903
2)12
+ 2 (1 minus 1199032)32
120587 (2 minus 1199032) + 4 (120587 minus 2) (1 minus 1199032)12
(47)
hold for all 119903 isin (0 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Natural Science Founda-tion of China under Grants 11371125 11401191 and 61374086the Natural Science Foundation of Zhejiang Province underGrant LY13A010004 the Natural Science Foundation ofHunan Province under Grant 12C0577 and the NaturalScience Foundation of the Zhejiang Broadcast and TV Uni-versity under Grant XKT-15G17
Journal of Function Spaces 5
References
[1] G Toader ldquoSome mean values related to the arithmetic-geometric meanrdquo Journal of Mathematical Analysis and Appli-cations vol 218 no 2 pp 358ndash368 1998
[2] E Neuman ldquoBounds for symmetric elliptic integralsrdquo Journalof Approximation Theory vol 122 no 2 pp 249ndash259 2003
[3] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegralsrdquo Journal of Approximation Theory vol 146 no 2 pp212ndash226 2007
[4] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegrals IIrdquo in Special Functions and Orthogonal PolynomialsContemporary Mathematics 471 pp 127ndash138 American Math-ematical Society Providence RI USA 2008
[5] Y-M Chu M-K Wang and X-Y Ma ldquoSharp bounds forToader mean in terms of contraharmonic mean with applica-tionsrdquo Journal of Mathematical Inequalities vol 7 no 2 pp 161ndash166 2013
[6] Y-Q Song W-D Jiang Y-M Chu and D-D Yan ldquoOptimalbounds for Toader mean in terms of arithmetic and contrahar-monic meansrdquo Journal of Mathematical Inequalities vol 7 no4 pp 751ndash757 2013
[7] W-H Li and M-M Zheng ldquoSome inequalities for boundingToader meanrdquo Journal of Function Spaces and Applications vol2013 Article ID 394194 5 pages 2013
[8] Y Hua and F Qi ldquoA double inequality for bounding Toadermean by the centroidal meanrdquo ProceedingsmdashMathematical Sci-ences vol 124 no 4 pp 527ndash531 2014
[9] Y Hua and F Qi ldquoThe best bounds for Toader mean in termsof the centroidal and arithmetic meansrdquo Filomat vol 28 no 4pp 775ndash780 2014
[10] MVuorinen ldquoHypergeometric functions in geometric functiontheoryrdquo in Special Functions andDifferential Equations (Madras1977) pp 119ndash126 Allied Publishers New Delhi India 1998
[11] S-L Qiu and J-M Shen ldquoOn two problems concerningmeansrdquoJournal of Hangzhou Institute of Electronic Engineering vol 17no 3 pp 1ndash7 1997 (Chinese)
[12] R W Barnard K Pearce and K C Richards ldquoAn inequalityinvolving the generalized hypergeometric function and the arclength of an ellipserdquo SIAM Journal on Mathematical Analysisvol 31 no 3 pp 693ndash699 2000
[13] H Alzer and S-L Qiu ldquoMonotonicity theorems and inequali-ties for the complete elliptic integralsrdquo Journal of Computationaland Applied Mathematics vol 172 no 2 pp 289ndash312 2004
[14] Y-M Chu M-K Wang S-L Qiu and Y-F Qiu ldquoSharpgeneralized Seiffert mean bounds for Toader meanrdquo Abstractand Applied Analysis vol 2011 Article ID 605259 8 pages 2011
[15] Y-M Chu and M-K Wang ldquoInequalities between arithmetic-geometric Gini and Toader meansrdquo Abstract and AppliedAnalysis vol 2012 Article ID 830585 11 pages 2012
[16] Y-M Chu and M-K Wang ldquoOptimal Lehmer mean boundsfor the Toader meanrdquo Results in Mathematics vol 61 no 3-4pp 223ndash229 2012
[17] G D Anderson M K Vamanamurthy and M K VuorinenConformal Invariants Inequalities and Quasiconformal MapsJohn Wiley amp Sons New York NY USA 1997
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2 Journal of Function Spaces
mean 119871119901(119886 119887) and 119902th generalized Seiffert mean 119878
119902(119886 119887) are
defined by
119872119901(119886 119887) =
(119886119901+ 119887119901
2)
1119901
119901 = 0
radic119886119887 119901 = 0
119866119901(119886 119887) =
(119886119901minus1
+ 119887119901minus1
119886 + 119887)
1(119901minus2)
119901 = 2
(119886119886119887119887)1(119886+119887)
119901 = 2
119871119901(119886 119887) =
119886119901+1
+ 119887119901+1
119886119901 + 119887119901
119878119902(119886 119887)
=
119901 (119886 minus 119887)
arctan [2119901 (119886 minus 119887) (119886 + 119887)] 0 lt 119901 le 1 119886 = 119887
119886 + 119887
2 119901 = 0 119886 = 119887
119886 119886 = 119887
(5)
respectively It is well known that119872119901(119886 119887)119866
119901(119886 119887) 119871
119901(119886 119887)
and 119878119902(119886 119887) are continuous and strictly increasing with
respect to 119901 isin R and 119902 isin [0 1] for fixed 119886 119887 gt 0 with 119886 = 119887respectively
Vuorinen [10] conjectured that inequality
11987232
(119886 119887) lt 119879 (119886 119887) (6)
holds for all 119886 119887 gt 0 with 119886 = 119887 This conjecture was provedby Qiu and Shen [11] and Barnard et al [12] respectively
Alzer and Qiu [13] presented a best possible upper powermean bound for the Toader mean as follows
119879 (119886 119887) lt 119872log 2(log120587minuslog 2) (119886 119887) (7)
for all 119886 119887 gt 0 with 119886 = 119887In [14 15] the authors found the best possible parameters
120572 120573 isin [0 1] and 120582 120583 isin R such that double inequalities119878120572(119886 119887) lt 119879(119886 119887) lt 119878
120573(119886 119887) and 119866
120582(119886 119887) lt 119879(119886 119887) lt
119866120583(119886 119887) hold for all 119886 119887 gt 0 with 119886 = 119887Chu and Wang [16] proved that double inequality
119871119901(119886 119887) lt 119879 (119886 119887) lt 119871
119902(119886 119887) (8)
holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 119901 le 0 and119902 ge 14
Inequality (8) leads to
119860 (119886 119887) = 1198710(119886 119887) lt 119879 (119886 119887) lt 119871
14(119886 119887) lt 119871
2(119886 119887)
= 119862 (119886 119887)
(9)
for all 119886 119887 gt 0 with 119886 = 119887Let 119886 119887 gt 0 with 119886 = 119887 be fixed and 119891(119909) = 119862[119909119886 + (1 minus
119909)119887 119909119887 + (1 minus 119909)119886] Then it is not difficult to verify that 119891(119909)is continuous and strictly increasing on [12 1] Note that
119891(1
2) = 119860 (119886 119887) lt 119879 (119886 119887) lt 119862 (119886 119887) = 119891 (1) (10)
Motivated by inequalities (9) and (10) it is natural to askwhat are the best possible parameters 120582
1 1205831isin R and 120582
2 1205832isin
(12 1) such that double inequalities
1205821119862 (119886 119887) + (1 minus 120582
1) 119860 (119886 119887) lt 119879 (119886 119887)
lt 1205831119862 (119886 119887) + (1 minus 120583
1) 119860 (119886 119887)
119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] lt 119879 (119886 119887)
lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886]
(11)
hold for all 119886 119887 gt 0 with 119886 = 119887 The main purpose of thispaper is to answer this question
2 Main Results
In order to prove our main results we need some basicknowledge and two lemmas which we present in this section
For 119903 isin (0 1) the complete elliptic integral K(119903) of thefirst kind is defined by
K (119903) = int
1205872
0
(1 minus 1199032sin2119905)
minus12
119889119905 (12)
We clearly see that
K (0+) = E (0
+) =
120587
2
E (1minus) = 1
(13)
and K(119903) and E(119903) satisfy formulas (see [17 Appendix E p474-475])
119889K (119903)
119889119903=E (119903) minus (1 minus 119903
2)K (119903)
119903 (1 minus 1199032)
119889E (119903)
119889119903=E (119903) minusK (119903)
119903
E(2radic119903
1 + 119903) =
2E (119903) minus (1 minus 1199032)K (119903)
1 + 119903
(14)
Lemma 1 (see [17 Theorem 125]) Let minusinfin lt 119886 lt 119887 lt
infin 119891 119892 [119886 119887] rarr (minusinfininfin) be continuous on [119886 119887] anddifferentiable on (119886 119887) and 119892
1015840(119909) = 0 on (119886 119887) If 1198911015840(119909)1198921015840(119909)
is increasing (decreasing) on (119886 119887) then so are
119891 (119909) minus 119891 (119886)
119892 (119909) minus 119892 (119886)
119891 (119909) minus 119891 (119887)
119892 (119909) minus 119892 (119887)
(15)
If 1198911015840(119909)1198921015840(119909) is strictly monotone then the monotonicity inthe conclusion is also strict
Lemma 2 (see [17 Theorem 321]) (1) Function 119903 997891rarr [E(119903) minus
(1 minus 1199032)K(119903)]119903
2 is strictly increasing from (0 1) to (1205874 1)(2) Function 119903 997891rarr (1 minus 119903
2)120582K(119903) is strictly decreasing from
(0 1) to (0 1205872) if 120582 ge 14
Journal of Function Spaces 3
Theorem 3 Double inequality
1205821119862 (119886 119887) + (1 minus 120582
1) 119860 (119886 119887) lt 119879 (119886 119887)
lt 1205831119862 (119886 119887) + (1 minus 120583
1) 119860 (119886 119887)
(16)
holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 1205821le 18 and
1205831ge 4120587 minus 1 = 02732
Proof Since 119860(119886 119887) 119879(119886 119887) and 119862(119886 119887) are symmetric andhomogeneous of degree 1 without loss of generality weassume that 119886 gt 119887 gt 0 Let 119903 = (119886 minus 119887)(119886 + 119887) isin (0 1)Then (1) and (2) lead to
119879 (119886 119887) =2119886
120587E(
2radic119903
1 + 119903)
=2119860 (119886 119887)
120587[2E (119903) minus (1 minus 119903
2)K (119903)]
(17)
119862 (119886 119887) = 119860 (119886 119887)1 + 3119903
2
1 + 1199032 (18)
We clearly see that inequality (16) is equivalent to
1205821lt
119879 (119886 119887) minus 119860 (119886 119887)
119862 (119886 119887) minus 119860 (119886 119887)lt 1205831 (19)
It follows from (17) and (18) that
119879 (119886 119887) minus 119860 (119886 119887)
119862 (119886 119887) minus 119860 (119886 119887)
=1
120587
2E (119903) minus (1 minus 1199032)K (119903) minus 1205872
1199032 (1 + 1199032)
(20)
Let
119891 (119903) =2E (119903) minus (1 minus 119903
2)K (119903) minus 1205872
1199032 (1 + 1199032)
1198911(119903) = 2E (119903) minus (1 minus 119903
2)K (119903) minus
120587
2
1198912(119903) =
1199032
1 + 1199032
(21)
Then simple computations lead to
1198911(0+) = 1198912(0+) = 0
1198911015840
1(119903)
11989110158402(119903)
=(1 + 119903
2)2
2
E (119903) minus (1 minus 1199032)K (119903)
1199032
(22)
From Lemmas 1 and 2 together with (21) and (22) weknow that 119891(119903) is strictly increasing on (0 1) and
119891 (0+) =
1198911015840
1(0+)
11989110158402(0+)
=120587
8
119891 (1minus) = 4 minus 120587
(23)
Therefore Theorem 3 follows from (19)ndash(21) and (23)together with the monotonicity of 119891(119903)
Theorem 4 Let 1205822 1205832isin (12 1) Then double inequality
119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] lt 119879 (119886 119887)
lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886]
(24)
holds for all 119886 119887 gt 0with 119886 = 119887 if and only if 1205822le 12+radic28 =
06767 and 1205832ge 12 + radic(4 minus 120587)(3120587 minus 4)2 = 06988
Proof Let 120582lowast
2= 12 + radic28 and 120583
lowast
2= 12 +
radic(4 minus 120587)(3120587 minus 4)2 We first prove that
119879 (119886 119887) gt 119862 [120582lowast
2119886 + (1 minus 120582
lowast
2) 119887 120582lowast
2119887 + (1 minus 120582
lowast
2) 119886] (25)
119879 (119886 119887) lt 119862 [120583lowast
2119886 + (1 minus 120583
lowast
2) 119887 120583lowast
2119887 + (1 minus 120583
lowast
2) 119886] (26)
for all 119886 119887 gt 0 with 119886 = 119887Without loss of generality we assume that 119886 gt 119887 Let 119903 =
(119886 minus 119887)(119886 + 119887) isin (0 1) and 119901 isin (12 1) Then (2) leads to
119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]
= 119860 (119886 119887)3 (2119901 minus 1)
2
1199032+ 1
(2119901 minus 1)2
1199032 + 1
(27)
It follows from (17) and (27) that
119879 (119886 119887) minus 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]
= 119860 (119886 119887) [2
120587(2E (119903) minus (1 minus 119903
2)K (119903))
+2
(2119901 minus 1)2
1199032 + 1
minus 3]
(28)
Let
119892 (119903) =2
120587(2E (119903) minus (1 minus 119903
2)K (119903))
+2
(2119901 minus 1)2
1199032 + 1
minus 3
(29)
1198921(119903) =
1
1199031198921015840(119903) (30)
4 Journal of Function Spaces
Then making use of Lemma 2 and simple computations leadto
119892 (0) = 0 (31)
119892 (1) =4
120587+
2
(2119901 minus 1)2
+ 1
minus 3 (32)
1198921(119903) =
2
120587
E (119903) minus (1 minus 1199032)K (119903)
1199032
minus4 (2119901 minus 1)
2
[(2119901 minus 1)2
1199032 + 1]2
(33)
1198921(0) =
1
2minus 4 (2119901 minus 1)
2
(34)
1198921(1) =
2
120587minus
4 (2119901 minus 1)2
[(2119901 minus 1)2
+ 1]2 (35)
We divide the proof into two cases
Case 1 Consider 119901 = 120582lowast
2= 12 + radic28 Then (34) becomes
1198921(0) = 0 (36)
It follows from Lemma 2(1) and (33) together with (36) that
1198921(119903) gt 0 (37)
for all 119903 isin (0 1)Therefore inequality (25) follows easily from (28)ndash(31)
and (37)
Case 2 Consider 119901 = 120583lowast
2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then
(32) (34) and (35) lead to
119892 (1) = 0 (38)
1198921(0) = minus
36 minus 11120587
6120587 minus 8lt 0 (39)
1198921(1) =
31205872minus 14120587 + 16
1205872gt 0 (40)
It follows from Lemma 2(1) (33) (39) and (40) that thereexists 119903
0isin (0 1) such that 119892
1(119903) lt 0 for 119903 isin (0 119903
0) and
1198921(119903) gt 0 for 119903 isin (119903
0 1)Then (30) leads to the conclusion that
119892(119903) is strictly decreasing on (0 1199030] and strictly increasing on
(1199030 1]Therefore inequality (26) follows easily from (28) (29)
(31) (38) and the piecewise monotonicity of 119892(119903)Next we prove that 120582
2= 120582lowast
2= 12 + radic28 is the best
possible parameter on (12 1) such that inequality
119879 (119886 119887) gt 119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] (41)
holds for all 119886 119887 gt 0 with 119886 = 119887Indeed if 120582lowast
2= 12 + radic28 lt 119901 lt 1 then (34) leads to
1198921(0) lt 0 and there exists 120575
1isin (0 1) such that
1198921(119903) lt 0 (42)
for 119903 isin (0 1205751)
Equations (28)ndash(31) and inequality (42) imply that
119879 (119886 119887) lt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (43)
for (119886 minus 119887)(119886 + 119887) isin (0 1205751)
Finally we prove that 1205832= 120583lowast
2= 12+radic(4 minus 120587)(3120587 minus 4)
2 is the best possible parameter on (12 1) such that doubleinequality
119879 (119886 119887) lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886] (44)
for all 119886 119887 gt 0 with 119886 = 119887In fact if 12 lt 119901 lt 120583
lowast
2= 12+radic(4 minus 120587)(3120587 minus 4)2 then
(32) leads to 119892(1) gt 0 and there exists 1205752isin (0 1) such that
119892 (119903) gt 0 (45)
for 119903 isin (1 minus 1205752 1)
Equations (28) and (29) together with inequality (45)imply that
119879 (119886 119887) gt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (46)
for (119886 minus 119887)(119886 + 119887) isin (1 minus 1205752 1)
Let 119903 isin (0 1) 119886 = 1 119887 = radic1 minus 1199032 1205821= 18 120583
1= 4120587 minus 1
1205822= 12 + radic28 and 120583
2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then
Theorems 3 and 4 lead to Corollary 5 as follows
Corollary 5 Double inequalities
120587 [1 + (1 minus 1199032)32
]
16 (2 minus 1199032)+
7120587 [1 + (1 minus 1199032)12
]
32lt E (119903)
lt
(4 minus 120587) [1 + (1 minus 1199032)32
]
2 (2 minus 1199032)+ (
120587
2minus 1) [1 + (1 minus 119903
2)12
]
120587
2
32 minus 211199032+ 21 (1 minus 119903
2)12
+ 11 (1 minus 1199032)32
36 minus 181199032 + 28 (1 minus 1199032)12
lt E (119903) lt120587
2
sdot3120587 minus 4 minus 3 (120587 minus 2) 119903
2+ 3 (120587 minus 2) (1 minus 119903
2)12
+ 2 (1 minus 1199032)32
120587 (2 minus 1199032) + 4 (120587 minus 2) (1 minus 1199032)12
(47)
hold for all 119903 isin (0 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Natural Science Founda-tion of China under Grants 11371125 11401191 and 61374086the Natural Science Foundation of Zhejiang Province underGrant LY13A010004 the Natural Science Foundation ofHunan Province under Grant 12C0577 and the NaturalScience Foundation of the Zhejiang Broadcast and TV Uni-versity under Grant XKT-15G17
Journal of Function Spaces 5
References
[1] G Toader ldquoSome mean values related to the arithmetic-geometric meanrdquo Journal of Mathematical Analysis and Appli-cations vol 218 no 2 pp 358ndash368 1998
[2] E Neuman ldquoBounds for symmetric elliptic integralsrdquo Journalof Approximation Theory vol 122 no 2 pp 249ndash259 2003
[3] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegralsrdquo Journal of Approximation Theory vol 146 no 2 pp212ndash226 2007
[4] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegrals IIrdquo in Special Functions and Orthogonal PolynomialsContemporary Mathematics 471 pp 127ndash138 American Math-ematical Society Providence RI USA 2008
[5] Y-M Chu M-K Wang and X-Y Ma ldquoSharp bounds forToader mean in terms of contraharmonic mean with applica-tionsrdquo Journal of Mathematical Inequalities vol 7 no 2 pp 161ndash166 2013
[6] Y-Q Song W-D Jiang Y-M Chu and D-D Yan ldquoOptimalbounds for Toader mean in terms of arithmetic and contrahar-monic meansrdquo Journal of Mathematical Inequalities vol 7 no4 pp 751ndash757 2013
[7] W-H Li and M-M Zheng ldquoSome inequalities for boundingToader meanrdquo Journal of Function Spaces and Applications vol2013 Article ID 394194 5 pages 2013
[8] Y Hua and F Qi ldquoA double inequality for bounding Toadermean by the centroidal meanrdquo ProceedingsmdashMathematical Sci-ences vol 124 no 4 pp 527ndash531 2014
[9] Y Hua and F Qi ldquoThe best bounds for Toader mean in termsof the centroidal and arithmetic meansrdquo Filomat vol 28 no 4pp 775ndash780 2014
[10] MVuorinen ldquoHypergeometric functions in geometric functiontheoryrdquo in Special Functions andDifferential Equations (Madras1977) pp 119ndash126 Allied Publishers New Delhi India 1998
[11] S-L Qiu and J-M Shen ldquoOn two problems concerningmeansrdquoJournal of Hangzhou Institute of Electronic Engineering vol 17no 3 pp 1ndash7 1997 (Chinese)
[12] R W Barnard K Pearce and K C Richards ldquoAn inequalityinvolving the generalized hypergeometric function and the arclength of an ellipserdquo SIAM Journal on Mathematical Analysisvol 31 no 3 pp 693ndash699 2000
[13] H Alzer and S-L Qiu ldquoMonotonicity theorems and inequali-ties for the complete elliptic integralsrdquo Journal of Computationaland Applied Mathematics vol 172 no 2 pp 289ndash312 2004
[14] Y-M Chu M-K Wang S-L Qiu and Y-F Qiu ldquoSharpgeneralized Seiffert mean bounds for Toader meanrdquo Abstractand Applied Analysis vol 2011 Article ID 605259 8 pages 2011
[15] Y-M Chu and M-K Wang ldquoInequalities between arithmetic-geometric Gini and Toader meansrdquo Abstract and AppliedAnalysis vol 2012 Article ID 830585 11 pages 2012
[16] Y-M Chu and M-K Wang ldquoOptimal Lehmer mean boundsfor the Toader meanrdquo Results in Mathematics vol 61 no 3-4pp 223ndash229 2012
[17] G D Anderson M K Vamanamurthy and M K VuorinenConformal Invariants Inequalities and Quasiconformal MapsJohn Wiley amp Sons New York NY USA 1997
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
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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
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 3
Theorem 3 Double inequality
1205821119862 (119886 119887) + (1 minus 120582
1) 119860 (119886 119887) lt 119879 (119886 119887)
lt 1205831119862 (119886 119887) + (1 minus 120583
1) 119860 (119886 119887)
(16)
holds for all 119886 119887 gt 0 with 119886 = 119887 if and only if 1205821le 18 and
1205831ge 4120587 minus 1 = 02732
Proof Since 119860(119886 119887) 119879(119886 119887) and 119862(119886 119887) are symmetric andhomogeneous of degree 1 without loss of generality weassume that 119886 gt 119887 gt 0 Let 119903 = (119886 minus 119887)(119886 + 119887) isin (0 1)Then (1) and (2) lead to
119879 (119886 119887) =2119886
120587E(
2radic119903
1 + 119903)
=2119860 (119886 119887)
120587[2E (119903) minus (1 minus 119903
2)K (119903)]
(17)
119862 (119886 119887) = 119860 (119886 119887)1 + 3119903
2
1 + 1199032 (18)
We clearly see that inequality (16) is equivalent to
1205821lt
119879 (119886 119887) minus 119860 (119886 119887)
119862 (119886 119887) minus 119860 (119886 119887)lt 1205831 (19)
It follows from (17) and (18) that
119879 (119886 119887) minus 119860 (119886 119887)
119862 (119886 119887) minus 119860 (119886 119887)
=1
120587
2E (119903) minus (1 minus 1199032)K (119903) minus 1205872
1199032 (1 + 1199032)
(20)
Let
119891 (119903) =2E (119903) minus (1 minus 119903
2)K (119903) minus 1205872
1199032 (1 + 1199032)
1198911(119903) = 2E (119903) minus (1 minus 119903
2)K (119903) minus
120587
2
1198912(119903) =
1199032
1 + 1199032
(21)
Then simple computations lead to
1198911(0+) = 1198912(0+) = 0
1198911015840
1(119903)
11989110158402(119903)
=(1 + 119903
2)2
2
E (119903) minus (1 minus 1199032)K (119903)
1199032
(22)
From Lemmas 1 and 2 together with (21) and (22) weknow that 119891(119903) is strictly increasing on (0 1) and
119891 (0+) =
1198911015840
1(0+)
11989110158402(0+)
=120587
8
119891 (1minus) = 4 minus 120587
(23)
Therefore Theorem 3 follows from (19)ndash(21) and (23)together with the monotonicity of 119891(119903)
Theorem 4 Let 1205822 1205832isin (12 1) Then double inequality
119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] lt 119879 (119886 119887)
lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886]
(24)
holds for all 119886 119887 gt 0with 119886 = 119887 if and only if 1205822le 12+radic28 =
06767 and 1205832ge 12 + radic(4 minus 120587)(3120587 minus 4)2 = 06988
Proof Let 120582lowast
2= 12 + radic28 and 120583
lowast
2= 12 +
radic(4 minus 120587)(3120587 minus 4)2 We first prove that
119879 (119886 119887) gt 119862 [120582lowast
2119886 + (1 minus 120582
lowast
2) 119887 120582lowast
2119887 + (1 minus 120582
lowast
2) 119886] (25)
119879 (119886 119887) lt 119862 [120583lowast
2119886 + (1 minus 120583
lowast
2) 119887 120583lowast
2119887 + (1 minus 120583
lowast
2) 119886] (26)
for all 119886 119887 gt 0 with 119886 = 119887Without loss of generality we assume that 119886 gt 119887 Let 119903 =
(119886 minus 119887)(119886 + 119887) isin (0 1) and 119901 isin (12 1) Then (2) leads to
119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]
= 119860 (119886 119887)3 (2119901 minus 1)
2
1199032+ 1
(2119901 minus 1)2
1199032 + 1
(27)
It follows from (17) and (27) that
119879 (119886 119887) minus 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886]
= 119860 (119886 119887) [2
120587(2E (119903) minus (1 minus 119903
2)K (119903))
+2
(2119901 minus 1)2
1199032 + 1
minus 3]
(28)
Let
119892 (119903) =2
120587(2E (119903) minus (1 minus 119903
2)K (119903))
+2
(2119901 minus 1)2
1199032 + 1
minus 3
(29)
1198921(119903) =
1
1199031198921015840(119903) (30)
4 Journal of Function Spaces
Then making use of Lemma 2 and simple computations leadto
119892 (0) = 0 (31)
119892 (1) =4
120587+
2
(2119901 minus 1)2
+ 1
minus 3 (32)
1198921(119903) =
2
120587
E (119903) minus (1 minus 1199032)K (119903)
1199032
minus4 (2119901 minus 1)
2
[(2119901 minus 1)2
1199032 + 1]2
(33)
1198921(0) =
1
2minus 4 (2119901 minus 1)
2
(34)
1198921(1) =
2
120587minus
4 (2119901 minus 1)2
[(2119901 minus 1)2
+ 1]2 (35)
We divide the proof into two cases
Case 1 Consider 119901 = 120582lowast
2= 12 + radic28 Then (34) becomes
1198921(0) = 0 (36)
It follows from Lemma 2(1) and (33) together with (36) that
1198921(119903) gt 0 (37)
for all 119903 isin (0 1)Therefore inequality (25) follows easily from (28)ndash(31)
and (37)
Case 2 Consider 119901 = 120583lowast
2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then
(32) (34) and (35) lead to
119892 (1) = 0 (38)
1198921(0) = minus
36 minus 11120587
6120587 minus 8lt 0 (39)
1198921(1) =
31205872minus 14120587 + 16
1205872gt 0 (40)
It follows from Lemma 2(1) (33) (39) and (40) that thereexists 119903
0isin (0 1) such that 119892
1(119903) lt 0 for 119903 isin (0 119903
0) and
1198921(119903) gt 0 for 119903 isin (119903
0 1)Then (30) leads to the conclusion that
119892(119903) is strictly decreasing on (0 1199030] and strictly increasing on
(1199030 1]Therefore inequality (26) follows easily from (28) (29)
(31) (38) and the piecewise monotonicity of 119892(119903)Next we prove that 120582
2= 120582lowast
2= 12 + radic28 is the best
possible parameter on (12 1) such that inequality
119879 (119886 119887) gt 119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] (41)
holds for all 119886 119887 gt 0 with 119886 = 119887Indeed if 120582lowast
2= 12 + radic28 lt 119901 lt 1 then (34) leads to
1198921(0) lt 0 and there exists 120575
1isin (0 1) such that
1198921(119903) lt 0 (42)
for 119903 isin (0 1205751)
Equations (28)ndash(31) and inequality (42) imply that
119879 (119886 119887) lt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (43)
for (119886 minus 119887)(119886 + 119887) isin (0 1205751)
Finally we prove that 1205832= 120583lowast
2= 12+radic(4 minus 120587)(3120587 minus 4)
2 is the best possible parameter on (12 1) such that doubleinequality
119879 (119886 119887) lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886] (44)
for all 119886 119887 gt 0 with 119886 = 119887In fact if 12 lt 119901 lt 120583
lowast
2= 12+radic(4 minus 120587)(3120587 minus 4)2 then
(32) leads to 119892(1) gt 0 and there exists 1205752isin (0 1) such that
119892 (119903) gt 0 (45)
for 119903 isin (1 minus 1205752 1)
Equations (28) and (29) together with inequality (45)imply that
119879 (119886 119887) gt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (46)
for (119886 minus 119887)(119886 + 119887) isin (1 minus 1205752 1)
Let 119903 isin (0 1) 119886 = 1 119887 = radic1 minus 1199032 1205821= 18 120583
1= 4120587 minus 1
1205822= 12 + radic28 and 120583
2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then
Theorems 3 and 4 lead to Corollary 5 as follows
Corollary 5 Double inequalities
120587 [1 + (1 minus 1199032)32
]
16 (2 minus 1199032)+
7120587 [1 + (1 minus 1199032)12
]
32lt E (119903)
lt
(4 minus 120587) [1 + (1 minus 1199032)32
]
2 (2 minus 1199032)+ (
120587
2minus 1) [1 + (1 minus 119903
2)12
]
120587
2
32 minus 211199032+ 21 (1 minus 119903
2)12
+ 11 (1 minus 1199032)32
36 minus 181199032 + 28 (1 minus 1199032)12
lt E (119903) lt120587
2
sdot3120587 minus 4 minus 3 (120587 minus 2) 119903
2+ 3 (120587 minus 2) (1 minus 119903
2)12
+ 2 (1 minus 1199032)32
120587 (2 minus 1199032) + 4 (120587 minus 2) (1 minus 1199032)12
(47)
hold for all 119903 isin (0 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Natural Science Founda-tion of China under Grants 11371125 11401191 and 61374086the Natural Science Foundation of Zhejiang Province underGrant LY13A010004 the Natural Science Foundation ofHunan Province under Grant 12C0577 and the NaturalScience Foundation of the Zhejiang Broadcast and TV Uni-versity under Grant XKT-15G17
Journal of Function Spaces 5
References
[1] G Toader ldquoSome mean values related to the arithmetic-geometric meanrdquo Journal of Mathematical Analysis and Appli-cations vol 218 no 2 pp 358ndash368 1998
[2] E Neuman ldquoBounds for symmetric elliptic integralsrdquo Journalof Approximation Theory vol 122 no 2 pp 249ndash259 2003
[3] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegralsrdquo Journal of Approximation Theory vol 146 no 2 pp212ndash226 2007
[4] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegrals IIrdquo in Special Functions and Orthogonal PolynomialsContemporary Mathematics 471 pp 127ndash138 American Math-ematical Society Providence RI USA 2008
[5] Y-M Chu M-K Wang and X-Y Ma ldquoSharp bounds forToader mean in terms of contraharmonic mean with applica-tionsrdquo Journal of Mathematical Inequalities vol 7 no 2 pp 161ndash166 2013
[6] Y-Q Song W-D Jiang Y-M Chu and D-D Yan ldquoOptimalbounds for Toader mean in terms of arithmetic and contrahar-monic meansrdquo Journal of Mathematical Inequalities vol 7 no4 pp 751ndash757 2013
[7] W-H Li and M-M Zheng ldquoSome inequalities for boundingToader meanrdquo Journal of Function Spaces and Applications vol2013 Article ID 394194 5 pages 2013
[8] Y Hua and F Qi ldquoA double inequality for bounding Toadermean by the centroidal meanrdquo ProceedingsmdashMathematical Sci-ences vol 124 no 4 pp 527ndash531 2014
[9] Y Hua and F Qi ldquoThe best bounds for Toader mean in termsof the centroidal and arithmetic meansrdquo Filomat vol 28 no 4pp 775ndash780 2014
[10] MVuorinen ldquoHypergeometric functions in geometric functiontheoryrdquo in Special Functions andDifferential Equations (Madras1977) pp 119ndash126 Allied Publishers New Delhi India 1998
[11] S-L Qiu and J-M Shen ldquoOn two problems concerningmeansrdquoJournal of Hangzhou Institute of Electronic Engineering vol 17no 3 pp 1ndash7 1997 (Chinese)
[12] R W Barnard K Pearce and K C Richards ldquoAn inequalityinvolving the generalized hypergeometric function and the arclength of an ellipserdquo SIAM Journal on Mathematical Analysisvol 31 no 3 pp 693ndash699 2000
[13] H Alzer and S-L Qiu ldquoMonotonicity theorems and inequali-ties for the complete elliptic integralsrdquo Journal of Computationaland Applied Mathematics vol 172 no 2 pp 289ndash312 2004
[14] Y-M Chu M-K Wang S-L Qiu and Y-F Qiu ldquoSharpgeneralized Seiffert mean bounds for Toader meanrdquo Abstractand Applied Analysis vol 2011 Article ID 605259 8 pages 2011
[15] Y-M Chu and M-K Wang ldquoInequalities between arithmetic-geometric Gini and Toader meansrdquo Abstract and AppliedAnalysis vol 2012 Article ID 830585 11 pages 2012
[16] Y-M Chu and M-K Wang ldquoOptimal Lehmer mean boundsfor the Toader meanrdquo Results in Mathematics vol 61 no 3-4pp 223ndash229 2012
[17] G D Anderson M K Vamanamurthy and M K VuorinenConformal Invariants Inequalities and Quasiconformal MapsJohn Wiley amp Sons New York NY USA 1997
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
Then making use of Lemma 2 and simple computations leadto
119892 (0) = 0 (31)
119892 (1) =4
120587+
2
(2119901 minus 1)2
+ 1
minus 3 (32)
1198921(119903) =
2
120587
E (119903) minus (1 minus 1199032)K (119903)
1199032
minus4 (2119901 minus 1)
2
[(2119901 minus 1)2
1199032 + 1]2
(33)
1198921(0) =
1
2minus 4 (2119901 minus 1)
2
(34)
1198921(1) =
2
120587minus
4 (2119901 minus 1)2
[(2119901 minus 1)2
+ 1]2 (35)
We divide the proof into two cases
Case 1 Consider 119901 = 120582lowast
2= 12 + radic28 Then (34) becomes
1198921(0) = 0 (36)
It follows from Lemma 2(1) and (33) together with (36) that
1198921(119903) gt 0 (37)
for all 119903 isin (0 1)Therefore inequality (25) follows easily from (28)ndash(31)
and (37)
Case 2 Consider 119901 = 120583lowast
2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then
(32) (34) and (35) lead to
119892 (1) = 0 (38)
1198921(0) = minus
36 minus 11120587
6120587 minus 8lt 0 (39)
1198921(1) =
31205872minus 14120587 + 16
1205872gt 0 (40)
It follows from Lemma 2(1) (33) (39) and (40) that thereexists 119903
0isin (0 1) such that 119892
1(119903) lt 0 for 119903 isin (0 119903
0) and
1198921(119903) gt 0 for 119903 isin (119903
0 1)Then (30) leads to the conclusion that
119892(119903) is strictly decreasing on (0 1199030] and strictly increasing on
(1199030 1]Therefore inequality (26) follows easily from (28) (29)
(31) (38) and the piecewise monotonicity of 119892(119903)Next we prove that 120582
2= 120582lowast
2= 12 + radic28 is the best
possible parameter on (12 1) such that inequality
119879 (119886 119887) gt 119862 [1205822119886 + (1 minus 120582
2) 119887 1205822119887 + (1 minus 120582
2) 119886] (41)
holds for all 119886 119887 gt 0 with 119886 = 119887Indeed if 120582lowast
2= 12 + radic28 lt 119901 lt 1 then (34) leads to
1198921(0) lt 0 and there exists 120575
1isin (0 1) such that
1198921(119903) lt 0 (42)
for 119903 isin (0 1205751)
Equations (28)ndash(31) and inequality (42) imply that
119879 (119886 119887) lt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (43)
for (119886 minus 119887)(119886 + 119887) isin (0 1205751)
Finally we prove that 1205832= 120583lowast
2= 12+radic(4 minus 120587)(3120587 minus 4)
2 is the best possible parameter on (12 1) such that doubleinequality
119879 (119886 119887) lt 119862 [1205832119886 + (1 minus 120583
2) 119887 1205832119887 + (1 minus 120583
2) 119886] (44)
for all 119886 119887 gt 0 with 119886 = 119887In fact if 12 lt 119901 lt 120583
lowast
2= 12+radic(4 minus 120587)(3120587 minus 4)2 then
(32) leads to 119892(1) gt 0 and there exists 1205752isin (0 1) such that
119892 (119903) gt 0 (45)
for 119903 isin (1 minus 1205752 1)
Equations (28) and (29) together with inequality (45)imply that
119879 (119886 119887) gt 119862 [119901119886 + (1 minus 119901) 119887 119901119887 + (1 minus 119901) 119886] (46)
for (119886 minus 119887)(119886 + 119887) isin (1 minus 1205752 1)
Let 119903 isin (0 1) 119886 = 1 119887 = radic1 minus 1199032 1205821= 18 120583
1= 4120587 minus 1
1205822= 12 + radic28 and 120583
2= 12 + radic(4 minus 120587)(3120587 minus 4)2 Then
Theorems 3 and 4 lead to Corollary 5 as follows
Corollary 5 Double inequalities
120587 [1 + (1 minus 1199032)32
]
16 (2 minus 1199032)+
7120587 [1 + (1 minus 1199032)12
]
32lt E (119903)
lt
(4 minus 120587) [1 + (1 minus 1199032)32
]
2 (2 minus 1199032)+ (
120587
2minus 1) [1 + (1 minus 119903
2)12
]
120587
2
32 minus 211199032+ 21 (1 minus 119903
2)12
+ 11 (1 minus 1199032)32
36 minus 181199032 + 28 (1 minus 1199032)12
lt E (119903) lt120587
2
sdot3120587 minus 4 minus 3 (120587 minus 2) 119903
2+ 3 (120587 minus 2) (1 minus 119903
2)12
+ 2 (1 minus 1199032)32
120587 (2 minus 1199032) + 4 (120587 minus 2) (1 minus 1199032)12
(47)
hold for all 119903 isin (0 1)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This research was supported by the Natural Science Founda-tion of China under Grants 11371125 11401191 and 61374086the Natural Science Foundation of Zhejiang Province underGrant LY13A010004 the Natural Science Foundation ofHunan Province under Grant 12C0577 and the NaturalScience Foundation of the Zhejiang Broadcast and TV Uni-versity under Grant XKT-15G17
Journal of Function Spaces 5
References
[1] G Toader ldquoSome mean values related to the arithmetic-geometric meanrdquo Journal of Mathematical Analysis and Appli-cations vol 218 no 2 pp 358ndash368 1998
[2] E Neuman ldquoBounds for symmetric elliptic integralsrdquo Journalof Approximation Theory vol 122 no 2 pp 249ndash259 2003
[3] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegralsrdquo Journal of Approximation Theory vol 146 no 2 pp212ndash226 2007
[4] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegrals IIrdquo in Special Functions and Orthogonal PolynomialsContemporary Mathematics 471 pp 127ndash138 American Math-ematical Society Providence RI USA 2008
[5] Y-M Chu M-K Wang and X-Y Ma ldquoSharp bounds forToader mean in terms of contraharmonic mean with applica-tionsrdquo Journal of Mathematical Inequalities vol 7 no 2 pp 161ndash166 2013
[6] Y-Q Song W-D Jiang Y-M Chu and D-D Yan ldquoOptimalbounds for Toader mean in terms of arithmetic and contrahar-monic meansrdquo Journal of Mathematical Inequalities vol 7 no4 pp 751ndash757 2013
[7] W-H Li and M-M Zheng ldquoSome inequalities for boundingToader meanrdquo Journal of Function Spaces and Applications vol2013 Article ID 394194 5 pages 2013
[8] Y Hua and F Qi ldquoA double inequality for bounding Toadermean by the centroidal meanrdquo ProceedingsmdashMathematical Sci-ences vol 124 no 4 pp 527ndash531 2014
[9] Y Hua and F Qi ldquoThe best bounds for Toader mean in termsof the centroidal and arithmetic meansrdquo Filomat vol 28 no 4pp 775ndash780 2014
[10] MVuorinen ldquoHypergeometric functions in geometric functiontheoryrdquo in Special Functions andDifferential Equations (Madras1977) pp 119ndash126 Allied Publishers New Delhi India 1998
[11] S-L Qiu and J-M Shen ldquoOn two problems concerningmeansrdquoJournal of Hangzhou Institute of Electronic Engineering vol 17no 3 pp 1ndash7 1997 (Chinese)
[12] R W Barnard K Pearce and K C Richards ldquoAn inequalityinvolving the generalized hypergeometric function and the arclength of an ellipserdquo SIAM Journal on Mathematical Analysisvol 31 no 3 pp 693ndash699 2000
[13] H Alzer and S-L Qiu ldquoMonotonicity theorems and inequali-ties for the complete elliptic integralsrdquo Journal of Computationaland Applied Mathematics vol 172 no 2 pp 289ndash312 2004
[14] Y-M Chu M-K Wang S-L Qiu and Y-F Qiu ldquoSharpgeneralized Seiffert mean bounds for Toader meanrdquo Abstractand Applied Analysis vol 2011 Article ID 605259 8 pages 2011
[15] Y-M Chu and M-K Wang ldquoInequalities between arithmetic-geometric Gini and Toader meansrdquo Abstract and AppliedAnalysis vol 2012 Article ID 830585 11 pages 2012
[16] Y-M Chu and M-K Wang ldquoOptimal Lehmer mean boundsfor the Toader meanrdquo Results in Mathematics vol 61 no 3-4pp 223ndash229 2012
[17] G D Anderson M K Vamanamurthy and M K VuorinenConformal Invariants Inequalities and Quasiconformal MapsJohn Wiley amp Sons New York NY USA 1997
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
References
[1] G Toader ldquoSome mean values related to the arithmetic-geometric meanrdquo Journal of Mathematical Analysis and Appli-cations vol 218 no 2 pp 358ndash368 1998
[2] E Neuman ldquoBounds for symmetric elliptic integralsrdquo Journalof Approximation Theory vol 122 no 2 pp 249ndash259 2003
[3] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegralsrdquo Journal of Approximation Theory vol 146 no 2 pp212ndash226 2007
[4] H Kazi and E Neuman ldquoInequalities and bounds for ellipticintegrals IIrdquo in Special Functions and Orthogonal PolynomialsContemporary Mathematics 471 pp 127ndash138 American Math-ematical Society Providence RI USA 2008
[5] Y-M Chu M-K Wang and X-Y Ma ldquoSharp bounds forToader mean in terms of contraharmonic mean with applica-tionsrdquo Journal of Mathematical Inequalities vol 7 no 2 pp 161ndash166 2013
[6] Y-Q Song W-D Jiang Y-M Chu and D-D Yan ldquoOptimalbounds for Toader mean in terms of arithmetic and contrahar-monic meansrdquo Journal of Mathematical Inequalities vol 7 no4 pp 751ndash757 2013
[7] W-H Li and M-M Zheng ldquoSome inequalities for boundingToader meanrdquo Journal of Function Spaces and Applications vol2013 Article ID 394194 5 pages 2013
[8] Y Hua and F Qi ldquoA double inequality for bounding Toadermean by the centroidal meanrdquo ProceedingsmdashMathematical Sci-ences vol 124 no 4 pp 527ndash531 2014
[9] Y Hua and F Qi ldquoThe best bounds for Toader mean in termsof the centroidal and arithmetic meansrdquo Filomat vol 28 no 4pp 775ndash780 2014
[10] MVuorinen ldquoHypergeometric functions in geometric functiontheoryrdquo in Special Functions andDifferential Equations (Madras1977) pp 119ndash126 Allied Publishers New Delhi India 1998
[11] S-L Qiu and J-M Shen ldquoOn two problems concerningmeansrdquoJournal of Hangzhou Institute of Electronic Engineering vol 17no 3 pp 1ndash7 1997 (Chinese)
[12] R W Barnard K Pearce and K C Richards ldquoAn inequalityinvolving the generalized hypergeometric function and the arclength of an ellipserdquo SIAM Journal on Mathematical Analysisvol 31 no 3 pp 693ndash699 2000
[13] H Alzer and S-L Qiu ldquoMonotonicity theorems and inequali-ties for the complete elliptic integralsrdquo Journal of Computationaland Applied Mathematics vol 172 no 2 pp 289ndash312 2004
[14] Y-M Chu M-K Wang S-L Qiu and Y-F Qiu ldquoSharpgeneralized Seiffert mean bounds for Toader meanrdquo Abstractand Applied Analysis vol 2011 Article ID 605259 8 pages 2011
[15] Y-M Chu and M-K Wang ldquoInequalities between arithmetic-geometric Gini and Toader meansrdquo Abstract and AppliedAnalysis vol 2012 Article ID 830585 11 pages 2012
[16] Y-M Chu and M-K Wang ldquoOptimal Lehmer mean boundsfor the Toader meanrdquo Results in Mathematics vol 61 no 3-4pp 223ndash229 2012
[17] G D Anderson M K Vamanamurthy and M K VuorinenConformal Invariants Inequalities and Quasiconformal MapsJohn Wiley amp Sons New York NY USA 1997
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