Abstract and Applied AnalysisVolume 2011 (2011), Article ID 671765, 8 pagesdoi:10.1155/2011/671765

Research Article

Monotonicity, Convexity, and Inequalities Involving the Agard Distortion Function

Yu-Ming Chu,¹ Miao-Kun Wang,¹ Yue-Ping Jiang,² and Song-Liang Qiu³

¹Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China²College of Mathematics and Econometrics, Hunan University, Changsha 410082, China³Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China

Received 22 June 2011; Accepted 10 November 2011

Academic Editor: Martin D. Schechter

Copyright © 2011 Yu-Ming Chu et al. 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.

Abstract.

We present some monotonicity, convexity, and inequalities for the Agard distortion function and improve some well-known results.

1. Introduction

For , Lengedre’s complete elliptic integrals of the first and second kind [1] are defined by respectively. Here and in what follows, we set .

Let be the modulus of the plan Grötzsch ring for , where is the unit disk. Then, it follows from [2] that

For , the Hersch-Pfluger distortion function is defined as while the Agard distortion function and the linear distortion function are defined by respectively.

It is well known that the functions and play a very important role in quasiconformal theory, quasiregular theory, and some other related fields [3–8]. For example, Martin [8] found that the sharp upper bound in Schottky’s theorem can be expressed by , and in [9–15] the authors established a number of remarkable properties for the Agard distortion function .

In [14], the authors proved that for all , where , . Recently, Anderson et al. [15] established that for all , where and are defined as in inequalities (1.6) and (1.7), respectively.

The purpose of this paper is to present the new monotonicity, convexity, and inequalities for the Agard distortion function and improve inequalities (1.6)–(1.9).

Our main results are Theorems 1.1 and 1.2 as follows.

Theorem 1.1. Let , , , and . Then, the following statements are true. (1) is strictly increasing from onto for ; if , then there exists , such that is strictly decreasing in and strictly increasing in . In particular, the inequality holds for all with the best possible constant .(2) is convex in for fixed .(3)If and , then is strictly increasing from onto .

Theorem 1.2. Let , , , , , and . Then, the following statements are true. (1) is strictly decreasing from onto for . If , then is strictly decreasing from onto . Moreover, for all and . In particular, if , then (1.10) becomes (2)If , then is strictly increasing from onto . Moreover, for all and . In particular, if , then (1.12) becomes (3)If , then there exists such that is strictly decreasing on and strictly increasing on .(4) is convex in .

2. Lemmas

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

The following formulas were presented in [14, Appendix  E, pp. 474-475]. Let ,, , and . Then,

Lemma 2.1 (see [14, Theorem  1.25]). For , let be continuous on and differentiable on , and let 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 can be found in [14, Theorem  3.21(1) and (7), Lemma  3.32(1) and Theorem  5.13(2)].

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

Lemma 2.3. Let , , and . Then, is strictly decreasing from onto .

Proof. Clearly , . Differentiating , one has where .From Lemma 2.2(1) and (2), we clearly see that is strictly decreasing in . Moreover, where is also strictly decreasing in . Thus, for , and for .Therefore, the monotonicity of follows from (2.3) and (2.4) together with the fact that for .

3. Proofs of Theorems 1.1 and 1.2

Proof of Theorem 1.1. For part (1), clearly . Let for , then , , Making use of (3.1), we have It follows from Lemma 2.2(3) that is strictly increasing from onto . Then, from (3.2) and (3.3), we know that is strictly increasing from onto for . If , then there exists such that is strictly decreasing in and strictly increasing in . Moreover, the inequality holds for all with the best possible constant .For part (2), denote . Differentiating , we get Let and , then , and Clearly, is strictly increasing in . Then, (3.5) and Lemma 2.1 lead to the conclusion that is strictly increasing in . Therefore, is convex in .For part (3), if , then . Let and , then , , and where is defined as in Lemma 2.2.Therefore, is strictly increasing in for follows from Lemmas 2.1 and 2.2 together with (3.6). Moreover, making use of l’Hôpital’s rule, we have , .

Proof of Theorem 1.2. Differentiating gives Let , and , then , , and Clearly, that is strictly increasing in follows from Lemma 2.2(1) and (2). Then, from (3.8) and (3.9) together with Lemma 2.1, we know that is strictly increasing in . Moreover, l’Hôpital’s rule leads to For part (1), if , then from (3.7) and (3.8), we know that for and is strictly decreasing in . Moreover, If , then is also strictly decreasing in and , and from Lemma 2.2(4) we get Therefore, inequalities (1.10) and (1.11) follows from (3.12) and the monotonicity of when .For part (2), if , then that is strictly increasing in follows from (3.7) and (3.8). Note that Therefore, inequalities (1.12) and (1.13) follow from (3.13) and the monotonicity of when .For part (3), if , then from (3.7) and (3.8) together with the monotonicity of we clearly see that there exists , such that for and for . Hence, is strictly decreasing in and strictly increasing in .Part (4) follows from (3.7) and (3.8) together with the monotonicity of .

Taking in Theorem 1.2, we get the following corollary.

Corollary 3.1. Let and be defined as in Theorem 1.2, , and . Then, (1)if , then is strictly decreasing from onto ; if , then is strictly decreasing from onto ;(2)if , then is strictly increasing from onto ;(3)if , then there exists , such that is strictly decreasing in and strictly increasing in ;(4) is convex in .

Inequalities (1.11) and (1.13) lead to the following corollary, which improve inequalities (1.6)–(1.9).

Corollary 3.2. Let and be defined as in Theorem 1.2, then the following inequality holds for all .

Acknowledgments

This research was supported by the Natural Science Foundation of China (Grants nos. 11071059, 11071069, and 11171307) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant no. T200924).

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