ISRN Mathematical AnalysisVolume 2011 (2011), Article ID 681702, 7 pagesdoi:10.5402/2011/681702

Research Article

On New Wilker-Type Inequalities

Zhengjie Sun and Ling Zhu

Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

Received 23 February 2011; Accepted 27 April 2011

Academic Editor: G. Lu

Copyright © 2011 Zhengjie Sun and Ling Zhu. 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.

In this paper, we present and prove a new Wilker-type inequality for hyperbolic functions. We also give a simple proof of the counterpart of the above inequality for the circular functions.

1. Introduction

Wilker [1] proposed two open questions as the following statements.

Problem 1. If , then

Problem 2. There exists a largest constant such that for .

Sumner et al. [2] affirmed the truth of the problems above and obtained a further result as follows.

Theorem 1.1. If , then Furthermore, and are the best constants in (1.3).

Guo et al. [3] gave new proofs of inequalities (1.1) and (1.2). In recent years, Zhu [4] showed a new simple proof of inequality (1.1); Pinelis [5] got other proof of inequalities (1.3) by using L'Hospital rules for monotonicity; Zhang and Zhu [6] gave a new elementary proof of double inequalities (1.3).

Besides, in article [7], Wang offered another type of the inequality and proved it by the power series expansions of circular functions as follows

Theorem 1.2. Let . Then, Furthermore, and are the best constants in (1.4).

Zhu [8] established a new Wilker-type inequality involving hyperbolic functions as follows: where the constant is the optimum constant in (1.5).

In fact, we can show a new Wilker-type inequality involving hyperbolic functions referring to the result above.

Theorem 1.3. Let . Then Furthermore, is the best constant in (1.6).

The purpose of this paper is to present a concise proof of inequality (1.6) by using the power series expansion of hyperbolic functions and to give an elementary proof of inequality (1.4) in another way.

2. The Proof of Theorem 1.3

Let , where ,  and . We get that the existence of Theorem 1.3 can be ensured when proving the following two statements: the first is or , and the second is .

Since where , , , and , according to the analysis above, the proof of Theorem 1.3 can be completed when proving that that is, for .

Let for . We compute where Then let ; we compute , where As we can see, all coefficients of the polynomial are positive integers. When , we have and . In view of that , we have that and for . We can also conclude that is increasing on , respectively. Since , we have for . So the proof of inequality (2.3) is completed. Furthermore, ; the proof of Theorem 1.3 is completed.

3. Another Proof of Theorem 1.2

We simplify the double inequality (1.4) into another form: Here we will discuss the monotonicity of the function . If the function is increasing for and both and are right, the proof of equality (1.4) is completed.

From [9] or [10, page 75], we know that for the equality holds. So we can also know that for hold. Then we compute by using the results above, where When , we can compute by using and . Furthermore, there is an obvious fact that for . So far, we can demonstrate that for all . So the function is increasing on . Evidently, and are true. So the proof of the double inequality of (1.4) is completed.

Remark 3.1. Wilker's inequalities (1.1) and (1.2) have been further refined by many scholars in the past few years; the readers can refer to [11–16].

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