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Research Article | Open Access

Volume 2011 |Article ID 697547 | https://doi.org/10.1155/2011/697547

Yu-Ming Chu, Miao-Kun Wang, Ye-Fang Qiu, "On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function", Abstract and Applied Analysis, vol. 2011, Article ID 697547, 7 pages, 2011. https://doi.org/10.1155/2011/697547

# On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

Accepted16 Aug 2011
Published19 Oct 2011

#### Abstract

We prove that the double inequality (( holds for all with the best possible constants and , which answer to an open problem proposed by Alzer and Qiu. Here, is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.

#### 1. Introduction

For , Lengedre's complete elliptic integrals of the first and second kind  are defined by respectively. Here and in what follows, we set . These integrals are special cases of Guassian hypergeometric function where . Indeed, we have

It is well known that the complete elliptic integrals have many important applications in physics, engineering, geometric function theory, quasiconformal analysis, theory of mean values, number theory, and other related fields .

Recently, the complete elliptic integrals have been the subject of intensive research. In particular, many remarkable properties and inequalities can be found in the literature [3, 1018].

In 1992, Anderson et al.  discovered that can be approximated by the inverse hyperbolic tangent function, , and proved that for .

In , Alzer and Qiu proved that the double inequality holds for all with the best possible constants and and proposed an open problem as follows.

Open Problem #
The double inequality holds for all with the best possible constants and .

It is the aim of this paper to give a positive answer to the open problem #.

#### 2. Lemmas and Theorem

In order to establish our main result, we need several formulas and lemmas, which we present in this section.

For , the following derivative formulas were presented in [4, Appendix  E, pages 474-475]:

Lemma 2.1 (see [4, Theorem  1.25]). For , let be continuous on and be differentiable on , let be on . If is increasing (decreasing) on , then so are If is strictly monotone, then the monotonicity in the conclusion is also strict.

The following Lemma 2.2 can be found in [9, Lemma  3(1)] and [4, Theorem  3.21(1) and Exercise 3.43(30) and (46)].

Lemma 2.2. (1)   is strictly decreasing in if and only if ;
(2)   is strictly increasing from onto ;
(3)   is strictly decreasing from onto ;
(4)   is strictly decreasing from onto .

Lemma 2.3. (1)   is strictly increasing from onto ;
(2)   is strictly increasing from onto ;
(3)   is strictly increasing from onto ;
(4)   is positive and strictly increasing in ;
(5)   is positive and strictly increasing on .

Proof. For part (1), let and . Then , and
It is well known that the function is strictly increasing from onto . Therefore, from (2.3) and Lemma 2.1 together with l'Hôpital's rule, we know that is strictly increasing in , and .
For part (2), clearly . Let and , then , , and
It follows from Lemma 2.1, Lemma 2.2(1), part (1), (2.4), and l'Hôpital's rule that is strictly increasing in and .
For part (3), from Lemma 2.2(4), we clearly see that . Let
From Lemma 2.2(2) and part (1), we clearly see that is strictly increasing in . Thus, the monotonicity of can be obtained from (2.5) and Lemma 2.1. Moreover, making use of l'Hôpital's rule, we have .
For part (4), let . Then, Lemma 2.2(3) leads to the conclusion that is strictly increasing in . Note that for .
Therefore, part (4) follows from (2.7) and (2.8).
For part (5), simple computations lead to
Making use of parts (1)–(4), one has for .
Therefore, part (5) follows from (2.9) and (2.11).

Lemma 2.4. Let then the following statements are true:
(1)   for all if and only if ;
(2)   for all if and only if .

Proof. Firstly, we prove that for . Since is continuous and strictly increasing with respect to for fixed , it suffices to prove that for all . Note that We divide the proof into two cases.
Case 1 (). Then, making use of Lemma 2.3(1)–(3) and (2.14), we have
Case 2 (). Then, making use of Lemma 2.3(4) and (2.14), we get
Inequalities (2.15) and (2.16) imply that is strictly increasing in . Therefore, follows from (2.13) and the monotonicity of .
On the other hand, inequality (1.5) leads to the conclusion that for all and .
Next, we prove that the parameters and 0 are the best possible parameters in Lemma 2.4(1) and (2), respectively.
If , then follows from Lemma 2.3(5). Moreover, let then, using l'Hôpital's rule and Lemma 2.2(4), we get
Inequality (2.18) implies that there exists such that for all . Therefore, for follows from (2.17).

From Lemma 2.4, we clearly see that the following Theorem 2.5 holds, which give a positive answer to the open problem #.

Theorem 2.5. The double inequality holds for all with the best possible constants and .

#### Acknowledgments

This paper was supported by the Natural Science Foundation of China under Grant 11071069 and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.

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