Jordan Type Inequalities for Hyperbolic Functions and Their Applications

Abstract

We present the best possible parameters p, such that the double inequality holds for all . As applications, some new inequalities for certain special function and bivariate means are found.

1. Introduction

The well known Jordan inequality [1] is given by

During the past few years, the improvements, refinements, and generalizations for inequality (1) have attracted the attention of many researchers [2–13]. Recently, the hyperbolic counterpart and its generalizations have been the subject of intensive research.

Zhu [14] proved that the inequality holds for all if and only if if .

In [3, 15], Neuman and Sándor proved that for all .

Klén et al. [5] proved that the double inequality holds for all .

In [4], the authors proved that the double inequality holds for all if and only if and .

Zhu [16, 17] proved that the inequalities hold for all if and only if ,   ,   ,   , and if ,   , and .

Very recently, Yang [18] proved that the double inequality holds for all if and only if and .

The main purpose of this paper is to find the best possible parameters such that the double inequality holds for all and present several new inequalities for certain special function and bivariate means.

2. Main Result

Theorem 1. Let . Then the double inequality holds for all if and only if and .

Proof. Let and let the function be defined on by Then making use of power series expansions and (9) we get Let Then for all . Consider for all .
It follows from (13) that for all .
Therefore, inequality (8) holds for all with and follows from (9)–(12) and (14).
Next, we prove that and if inequality (8) holds for all .
If the first inequality of (8) holds for all , then from (9) and (10) we have and .
If the second inequality of (8) holds for all , then it follows from (9) that
We clearly see that if . Therefore, follows from (16).

Remark 2. It is not difficult to verify that the bound given in Theorem 1 is strictly increasing with respect to on for fixed .

Remark 3. Let and . Then Theorem 1 and Remark 2 lead to for all .

3. Applications

It is well known that where is the trigamma function defined by

Let Then Remark 3 leads to for all .

From (19) and (22) we get the following.

Remark 4. For all one has In particular, we have
For , the Schwab-Borchardt mean [19–21] is given by
Let and let . Then and . It follows from Remark 3 and (26) that Note that
From (27) and (28) we get the following.

Remark 5. Let ; then the Schwab-Borchardt mean satisfies the double inequality
Let and let with . Then the arithmetic mean , logarithmic mean , geometric mean , and th power mean are defined by
It is well known that is continuous and strictly increasing with respect to for fixed with ; the main properties for the power mean are given in [22]. Recently, the arithmetic, logarithmic, geometric, and power means have been the subject of intensive research. In particular, many remarkable inequalities can be found in the literature [23–35].
Let ; then (30) leads to
From Theorem 1 and (31) we get the following.

Remark 6. Let ; then the double inequality holds for all with if and only if and . In particular, the double inequality holds for all with .

Conflict of Interests

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

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 61374086 and 11171307 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.