Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

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 inequalityholds 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 inequalityholds 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 inequalityfor 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 inequalityholds 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 inequalitieshold for all with , with and .

Gao et al. [15] provided the greatest value and the least value such that the double inequalityholds for all with , where is the first Seiffert mean of and .

Very recently, Yang [16] introduced the Yang meanof 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 inequalityholds 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 . Iffor andfor , 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 , andThen 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) becomesTherefore, part follows from (19).
For part , let be the unique solution of the equation on the interval , , , , , , , , , and . Then elaborated computations lead tofor .
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 towhere 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 thatfor 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) becomesTherefore, 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 getEquation (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 getEquation (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.