Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 8901258, 7 pages
http://dx.doi.org/10.1155/2016/8901258

Research Article

Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

Wei-Mao Qian¹ and Yu-Ming Chu²

¹School of Distance Education, Huzhou Broadcast and TV University, Huzhou 313000, China
²Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 23 January 2016; Revised 16 March 2016; Accepted 28 March 2016

Academic Editor: Kishin Sadarangani

Copyright © 2016 Wei-Mao Qian and Yu-Ming Chu. 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 prove that the double inequality holds for all with if and only if and and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, where is the unique solution of the equation on the interval , , and , and are the Yang, and th generalized logarithmic means of and , respectively.

1. Introduction

For and with , the th generalized logarithmic mean is defined by

It is well known that is continuous and strictly increasing with respect to for fixed with . Many classical bivariate means are the special case of the generalized logarithmic mean. For example, is the geometric mean, is the logarithmic mean, is the identric mean, and is the arithmetic mean. Recently, the generalized logarithmic mean has been the subject of intensive research.

Stolarsky [1] proved that the inequality

holds for all with and , and inequality (2) is reversed for , where and is the th power mean of and .

Yang [2] proved that the double inequality

holds for all with if , and inequality (3) is reversed if .

In [3], the authors proved that the inequality

holds for all with and .

Li et al. [4] proved that the function is strictly increasing (decreasing) on if (). In [5, 6], the authors proved that the function is strictly decreasing on if and the function is strictly increasing on for all .

Shi and Wu [7] proved that the double inequality

for all and if , and inequality (5) is reversed if .

Long and Chu [8] and Matejíčka [9] presented the best possible parameters and such that the double inequality

holds for all with and .

In [10], Qian and Long answered the question: what are the greatest value and the least value such that the double inequality

holds for all with and , where is the harmonic mean of and .

In [11, 12], the authors proved that the double inequalities

hold for all with if and only if , , , , where is the unique solution of the equation on the interval , is the unique solution of the equation on the interval , is the Neuman-Sándor mean, and is the second Seiffert mean.

In [13, 14], the authors presented the best possible parameters , , , and such that the double inequalities

hold for all with , with and .

Gao et al. [15] provided the greatest value and the least value such that the double inequality

holds for all with , where is the first Seiffert mean of and .

Very recently, Yang [16] introduced the Yang mean

of two distinct positive real numbers and and proved that the inequalities

hold for all with , where is the quadratic mean of and .

The Yang mean is the special case of the Seiffert type mean defined by Toader in [17], where is a bivariate mean and is a positive real number. Indeed, .

In [18, 19], the authors proved that the double inequalities

hold for all with if and only if , , , , , and , where is the unique solution of the equation on the interval , and .

Zhou et al. [20] proved that and are the best possible parameters such that the double inequality

holds for all with .

The main purpose of this paper is to present the best possible parameters and such that the double inequality holds for all with . As application, we derive several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions. Some complicated computations are carried out using Mathematica computer algebra system.

2. Lemmas

In order to prove our main result we need two lemmas, which we present in this section.

Lemma 1. Let , , be -times differentiable functions such that and for and . If

for and

for , then there exists such that for and for .

Proof. From (15) and (17) we clearly see that there exists such that for and for , which implies that is strictly decreasing on and strictly increasing on . Then (16) leads to the conclusion that there exists such that for and for .
Making use of (16) and the same method as above we know that for there exists such that for and for .

Lemma 2. Let , and

Then the following statements are true:

(1)if , then for all ;(2)if is the unique solution of the equation on the interval and , then there exists such that for and for .

Proof. For part , if , then (18) becomes

Therefore, part follows from (19).
For part , let be the unique solution of the equation on the interval , , , , , , , , , and . Then elaborated computations lead to

for .
Therefore, part follows easily from Lemma 1 and (20).

3. Main Result

Theorem 3. The double inequality

holds for all with if and only if and , where is the unique solution of the equation on the interval .

Proof. Since and are symmetric and homogeneous of degree one, without loss of generality, we assume that and . Let and . Then (1) and (11) lead to

where

where is defined by (15).
We divide the proof into four cases.
Case 1 . Then from Lemma 2 and (29) we clearly see that the function is strictly decreasing on . Then (27) leads to the conclusion that

for all . Therefore,

follows easily from (22), (23), (25), and (30).
Case 2 . Then from Lemma 2 and (29) we know that there exists such that the function is strictly increasing on and strictly decreasing on .
It follows from (25)–(28) and the piecewise monotonicity of the function that there exists such that the function is strictly increasing on and strictly decreasing on .
Note that (24) becomes

Therefore,

follows easily from (22), (23), and (32) together with the piecewise monotonicity of the function .
Case 3 . Let and ; then making use of Taylor expansion we get

Equation (34) implies that there exists small enough such that

for all .
Case 4 . Then from (24) and the fact that the function is strictly decreasing on we get

Equation (22) and inequality (36) imply that there exists large enough such that

for all .

4. Applications

As applications of Theorem 3 in engineering problems, we present several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions in this section.

From (1) and (11) together with Theorem 3 we get Theorem 4 immediately.

Theorem 4. Let be the unique solution of the equation on the interval . Then the double inequality

holds for all with .

Let , , and . Then Theorem 4 leads to the following.

Theorem 5. Let be the unique solution of the equation on the interval . Then the double inequality

holds for all .

Let , . Then (1) and (11) lead to

It follows from Theorem 3 and (40) that one has the following theorem.

Theorem 6. Let be the unique solution of the equation on the interval . Then the double inequality

holds for all .

Let , . Then (1) and (11) lead to

Theorem 3 and (42) lead to the following.

Theorem 7. Let be the unique solution of the equation on the interval . Then the double inequality

holds for all .

Let , , . Then from (1) and (11) we have

From (44), (45), and Theorem 3 one has the following.

Theorem 8. Let be the unique solution of the equation on the interval . Then the double inequalities

hold for all and .

Competing Interests

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

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grants 11371125, 61374086, and 11401191, the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT15G-17, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

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