Sharp Bounds for Toader Mean in terms of Arithmetic and Second Contraharmonic Means

Abstract

We present the best possible parameters [subscript]λ1[/subscript] ,[subscript]μ1[/subscript] ∈R and [subscript]λ2[/subscript] ,[subscript]μ2[/subscript] ∈(1/2,1) such that double inequalities [subscript]λ1[/subscript] C(a,b)+(1-[subscript]λ1[/subscript] )A(a,b)<T(a,b)<[subscript]μ1[/subscript] C(a,b)+(1-[subscript]μ1[/subscript] )A(a,b), C[[subscript]λ2[/subscript] a+(1-[subscript]λ2[/subscript] )b,[subscript]λ2[/subscript] b+(1-[subscript]λ2[/subscript] )a]<T(a,b)<C[[subscript]μ2[/subscript] a+(1-[subscript]μ2[/subscript] )b,[subscript]μ2[/subscript] b+(1-[subscript]μ2[/subscript] )a] hold for all a,b>0 with a≠b, where A(a,b)=(a+b)/2, C(a,b)=([superscript]a3[/superscript] +[superscript]b3[/superscript] )/([superscript]a2[/superscript] +[superscript]b2[/superscript] ) and T(a,b)=2[superscript]∫0π/2[/superscript] [superscript]a2[/superscript] [superscript]cos2[/superscript] θ+[superscript]b2[/superscript] [superscript]sin2[/superscript] θdθ/π are the arithmetic, second contraharmonic, and Toader means of a and b, respectively.

Wei-Mao Qian 1 and Ying-Qing Song 2 and Xiao-Hui Zhang 2 and Yu-Ming Chu 2

Academic Editor:Lars E. Persson

1, School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
2, School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China

Received 25 April 2015; Accepted 10 September 2015; 28 September 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

For [figure omitted; refer to PDF] the Toader mean [figure omitted; refer to PDF] [1], second contraharmonic mean [figure omitted; refer to PDF] , and arithmetic mean [figure omitted; refer to PDF] of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are given by [figure omitted; refer to PDF] respectively, where [figure omitted; refer to PDF] is the complete elliptic integral of the second kind. The Toader mean [figure omitted; refer to PDF] is well known in mathematical literature for many years; it satisfies [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] stands for the symmetric complete elliptic integral of the second kind (see [2-4]); therefore it cannot be expressed in terms of the elementary transcendental functions.

Recently, the Toader mean [figure omitted; refer to PDF] has been the subject of intensive research. In particular, many remarkable inequalities for the Toader mean can be found in the literature [5-9].

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then the [figure omitted; refer to PDF] th power mean [figure omitted; refer to PDF] , [figure omitted; refer to PDF] th Gini mean [figure omitted; refer to PDF] , [figure omitted; refer to PDF] th Lehmer mean [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] th generalized Seiffert mean [figure omitted; refer to PDF] are defined by [figure omitted; refer to PDF] respectively. It is well known that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are continuous and strictly increasing with respect to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for fixed [figure omitted; refer to PDF] with [figure omitted; refer to PDF] , respectively.

Vuorinen [10] conjectured that inequality [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] . This conjecture was proved by Qiu and Shen [11] and Barnard et al. [12], respectively.

Alzer and Qiu [13] presented a best possible upper power mean bound for the Toader mean as follows: [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

In [14, 15], the authors found the best possible parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that double inequalities [figure omitted; refer to PDF] and [figure omitted; refer to PDF] hold for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

Chu and Wang [16] proved that double inequality [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Inequality (8) leads to [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] with [figure omitted; refer to PDF] be fixed and [figure omitted; refer to PDF] . Then it is not difficult to verify that [figure omitted; refer to PDF] is continuous and strictly increasing on [figure omitted; refer to PDF] . Note that [figure omitted; refer to PDF]

Motivated by inequalities (9) and (10), it is natural to ask what are the best possible parameters, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , such that double inequalities [figure omitted; refer to PDF] hold for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] ? The main purpose of this paper is to answer this question.

2. Main Results

In order to prove our main results we need some basic knowledge and two lemmas, which we present in this section.

For [figure omitted; refer to PDF] the complete elliptic integral [figure omitted; refer to PDF] of the first kind is defined by [figure omitted; refer to PDF]

We clearly see that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfy formulas (see [17, Appendix E, p. 474-475]) [figure omitted; refer to PDF]

Lemma 1 (see [17, Theorem 1.25]).

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] be continuous on [figure omitted; refer to PDF] and differentiable on [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] is increasing (decreasing) on [figure omitted; refer to PDF] , then so are [figure omitted; refer to PDF] If [figure omitted; refer to PDF] is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2 (see [17, Theorem 3.21]).

(1) Function [figure omitted; refer to PDF] is strictly increasing from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] .

(2) Function [figure omitted; refer to PDF] is strictly decreasing from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] if [figure omitted; refer to PDF] .

Theorem 3.

Double inequality [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ....

Proof.

Since [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are symmetric and homogeneous of degree 1, without loss of generality, we assume that [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] . Then (1) and (2) lead to [figure omitted; refer to PDF]

We clearly see that inequality (16) is equivalent to [figure omitted; refer to PDF]

It follows from (17) and (18) that [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] Then simple computations lead to [figure omitted; refer to PDF]

From Lemmas 1 and 2 together with (21) and (22) we know that [figure omitted; refer to PDF] is strictly increasing on [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

Therefore, Theorem 3 follows from (19)-(21) and (23) together with the monotonicity of [figure omitted; refer to PDF] .

Theorem 4.

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Then double inequality [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] ... and [figure omitted; refer to PDF] ....

Proof.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . We first prove that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

Without loss of generality, we assume that [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then (2) leads to [figure omitted; refer to PDF]

It follows from (17) and (27) that [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] Then making use of Lemma 2 and simple computations lead to [figure omitted; refer to PDF]

We divide the proof into two cases.

Case 1. Consider [figure omitted; refer to PDF] . Then (34) becomes [figure omitted; refer to PDF] It follows from Lemma 2(1) and (33) together with (36) that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] .

Therefore, inequality (25) follows easily from (28)-(31) and (37).

Case 2. Consider [figure omitted; refer to PDF] . Then (32), (34), and (35) lead to [figure omitted; refer to PDF] It follows from Lemma 2(1), (33), (39), and (40) that there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Then (30) leads to the conclusion that [figure omitted; refer to PDF] is strictly decreasing on [figure omitted; refer to PDF] and strictly increasing on [figure omitted; refer to PDF] .

Therefore, inequality (26) follows easily from (28), (29), (31), (38), and the piecewise monotonicity of [figure omitted; refer to PDF] .

Next, we prove that [figure omitted; refer to PDF] is the best possible parameter on [figure omitted; refer to PDF] such that inequality [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

Indeed, if [figure omitted; refer to PDF] , then (34) leads to [figure omitted; refer to PDF] and there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Equations (28)-(31) and inequality (42) imply that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Finally, we prove that [figure omitted; refer to PDF] is the best possible parameter on [figure omitted; refer to PDF] such that double inequality [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

In fact, if [figure omitted; refer to PDF] , then (32) leads to [figure omitted; refer to PDF] and there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Equations (28) and (29) together with inequality (45) imply that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Then Theorems 3 and 4 lead to Corollary 5 as follows.

Corollary 5.

Double inequalities [figure omitted; refer to PDF] hold for all [figure omitted; refer to PDF] .

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11371125, 11401191, and 61374086, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, the Natural Science Foundation of Hunan Province under Grant 12C0577, and the Natural Science Foundation of the Zhejiang Broadcast and TV University under Grant XKT-15G17.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

[1] G. Toader, "Some mean values related to the arithmetic-geometric mean," Journal of Mathematical Analysis and Applications , vol. 218, no. 2, pp. 358-368, 1998.

[2] E. Neuman, "Bounds for symmetric elliptic integrals," Journal of Approximation Theory , vol. 122, no. 2, pp. 249-259, 2003.

[3] H. Kazi, E. Neuman, "Inequalities and bounds for elliptic integrals," Journal of Approximation Theory , vol. 146, no. 2, pp. 212-226, 2007.

[4] H. Kazi, E. Neuman, "Inequalities and bounds for elliptic integrals II," Special Functions and Orthogonal Polynomials , of Contemporary Mathematics 471, pp. 127-138, American Mathematical Society, Providence, RI, USA, 2008.

[5] Y.-M. Chu, M.-K. Wang, X.-Y. Ma, "Sharp bounds for Toader mean in terms of contraharmonic mean with applications," Journal of Mathematical Inequalities , vol. 7, no. 2, pp. 161-166, 2013.

[6] Y.-Q. Song, W.-D. Jiang, Y.-M. Chu, D.-D. Yan, "Optimal bounds for Toader mean in terms of arithmetic and contraharmonic means," Journal of Mathematical Inequalities , vol. 7, no. 4, pp. 751-757, 2013.

[7] W.-H. Li, M.-M. Zheng, "Some inequalities for bounding Toader mean," Journal of Function Spaces and Applications , vol. 2013, 2013.

[8] Y. Hua, F. Qi, "A double inequality for bounding Toader mean by the centroidal mean," Proceedings--Mathematical Sciences , vol. 124, no. 4, pp. 527-531, 2014.

[9] Y. Hua, F. Qi, "The best bounds for Toader mean in terms of the centroidal and arithmetic means," Filomat , vol. 28, no. 4, pp. 775-780, 2014.

[10] M. Vuorinen, "Hypergeometric functions in geometric function theory," Special Functions and Differential Equations (Madras, 1977) , pp. 119-126, Allied Publishers, New Delhi, India, 1998.

[11] S.-L. Qiu, J.-M. Shen, "On two problems concerning means," Journal of Hangzhou Institute of Electronic Engineering , vol. 17, no. 3, pp. 1-7, 1997.

[12] R. W. Barnard, K. Pearce, K. C. Richards, "An inequality involving the generalized hypergeometric function and the arc length of an ellipse," SIAM Journal on Mathematical Analysis , vol. 31, no. 3, pp. 693-699, 2000.

[13] H. Alzer, S.-L. Qiu, "Monotonicity theorems and inequalities for the complete elliptic integrals," Journal of Computational and Applied Mathematics , vol. 172, no. 2, pp. 289-312, 2004.

[14] Y.-M. Chu, M.-K. Wang, S.-L. Qiu, Y.-F. Qiu, "Sharp generalized Seiffert mean bounds for Toader mean," Abstract and Applied Analysis , vol. 2011, 2011.

[15] Y.-M. Chu, M.-K. Wang, "Inequalities between arithmetic-geometric, Gini, and Toader means," Abstract and Applied Analysis , vol. 2012, 2012.

[16] Y.-M. Chu, M.-K. Wang, "Optimal Lehmer mean bounds for the Toader mean," Results in Mathematics , vol. 61, no. 3-4, pp. 223-229, 2012.

[17] G. D. Anderson, M. K. Vamanamurthy, M. K. Vuorinen Conformal Invariants, Inequalities, and Quasiconformal Maps , John Wiley & Sons, New York, NY, USA, 1997.