Refinements of Bounds for Neuman Means

Abstract

We present the sharp bounds for the Neuman means , , and in terms of the arithmetic, harmonic, and contraharmonic means. Our results are the refinements or improvements of the results given by Neuman.

1. Introduction

For with , the Schwab-Borchardt mean of and is given by where and are the inverse cosine and inverse hyperbolic cosine functions, respectively.

It is well-known that the mean is strictly increasing in both and , nonsymmetric and homogeneous of degree with respect to and . Many symmetric bivariate means are special cases of the Schwab-Borchardt mean; for example, where , , and denote the classical geometric mean, arithmetic mean, and quadratic mean of and , respectively. The Schwab-Borchardt mean was investigated in [1, 2].

Let and be the harmonic and contraharmonic means of two positive numbers and , respectively. Then, it is well-known that for with .

Recently, the Schwab-Borchardt mean and its special cases have been the subject of intensive research. Neuman and Sándor [3, 4] proved that the inequalities hold for all , with . In [5], the author proved that the double inequalities hold for all , with if and only if , , , and . Chu and Long [6] found that the double inequality holds for all with if and only if and , where and is the th power mean of and . Zhao et al. [7] presented the least values , , and and the greatest values , , and such that the double inequalities hold for all , with .

Very recently, the bivariate means , , , and derived from the Schwab-Borchardt mean are defined by Neuman [8, 9] as follows:

We call the means , , , and given in (8) the Neuman means. Moreover, let ; then the following explicit formulas for , , , and are found by Neuman [8]: where , , , and are defined implicitly as , , , and , respectively. Clearly, , , , and .

In [8, 9], Neuman proved that the inequalities hold for , with .

He et al. [10] found the greatest values , , , , and the least values , , , such that the double inequalities hold for all with .

Motivated by inequalities (12), it is natural to ask what the greatest values , , , and and the least values , , , and are such that the double inequalities hold for all , with .

The purpose of this paper is to answer these questions. All numerical computations are carried out using MATHEMATICA software. Our main results are the following Theorems 1–4.

Theorem 1. The double inequality holds for all with if and only if and .

Theorem 2. The two-sided inequality holds true for all with if and only if and .

Theorem 3. The double inequality holds for all with if and only if and .

Theorem 4. The two-sided inequality holds true for all with if and only if and .

2. Two Lemmas

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

Lemma 5. Let and Then, the following statements are true.(1)If , then for all and for all .(2)If , then there exists such that for and for .(3)If , then there exists such that for and for .

Proof. For part (1), if , then (19) becomes Therefore, part (1) follows easily from (20).
For part (2), if , then simple computations lead to
It follows from (21)–(23) and (29) that is strictly increasing on . Then, (27) and (28) lead to the conclusion that there exists such that is strictly decreasing in and strictly increasing in .
Therefore, part (2) follows from (24) and (25) together with the piecewise monotonicity of .
For part (3), if , then numerical computations lead to
It follows from (26) and (30)–(32) that for .
Therefore, part (3) follows easily from (33)–(35).

Lemma 6. Let and Then, the following statements are true.(1)If , then for all and for all .(2)If , then there exists such that for and for .

Proof. For part (1), if , then (36) becomes Therefore, part (1) follows from (37).
For part (2), if , then numerical computations lead to
From (38) and (39) together with (42), we clearly see that for .
Therefore, part (2) follows from (40) and (41) together with (43).

3. Proofs of Theorems 1–4

Proof of Theorem 1. Without loss of generality, we assume that . Let ,  ,  , and . Then, , , , where where is defined as in Lemma 5.
We divide the proof into two cases.
Case  1  (). Then, from Lemma 5(1) and (49), we clearly see that is strictly decreasing in . Therefore, for all with follows from (45) and (48) together with the monotonicity of .
Case  2  (). Then, from (47) and (49) and Lemma 5(2), we know that and there exists such that is strictly decreasing in and strictly increasing in . Therefore, for all with follows from (45) and (48) together with (51) and the piecewise monotonicity of .
Note that
Therefore, Theorem 1 follows from (50) and (52)–(54) together with the following statements.(i)If , then (44) and (53) imply that there exists small enough such that for all with .(ii)If , then (44) and (54) imply that there exists large enough such that for all with .

Proof of Theorem 2. Without loss of generality, we assume that . Let ,  ,  , and . Then, ,  ,  , where where is defined as in Lemma 5.
We divide the proof into two cases.
Case  1 (). Then, from (59) and (60) together with Lemma 5(3), we clearly see that there exists such that is strictly increasing in and strictly decreasing in , and
Therefore, for all with follows easily from (56) and (58) together with (61) and the piecewise monotonicity of .
Case  2 (). Then, Lemma 5(1) and (60) lead to the conclusion that is strictly decreasing in . Therefore, for all with follows from (56) and (58) together with the monotonicity of .
Note that
Therefore, Theorem 2 follows from (55) and (62)–(65).

Proof of Theorem 3. Without loss of generality, we assume that . Let , , , and . Then, and (9) leads to
It follows from (66) that where where is defined as in Lemma 6.
If , then Lemma 6(1) and (71) lead to the conclusion that is strictly increasing in . Therefore, for all , with follows from (68) and (70) together with the monotonicity of .
Note that
Therefore, Theorem 3 follows from (12) and (67) together with (72)–(74).

Proof of Theorem 4. Without loss of generality, we assume that . Let ,  , , and . Then, ,  , and   and (10) leads to
It follows from (75) that where where is defined as in Lemma 6.
We divide the proof into two cases.
Case  1 (). Then, (81) and Lemma 6(1) lead to the conclusion that is strictly increasing in . Therefore, for all with follows easily from (77) and (79) together with the monotonicity of .
Case  2 (). Then, (80) and (81) together with Lemma 6(2) lead to the conclusion that there exists such that is strictly decreasing in and strictly increasing in , and Therefore, for all with follows easily from (77) and (79) together with (83) and the piecewise monotonicity of .
Note that
Therefore, Theorem 4 follows from (76) and (82) together with (84)–(86).

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, the Natural Science Foundation of the Open University of China under Grant Q1601E-Y, and the Natural Science Foundation of Zhejiang Broadcast and TV University under Grant XKT-13Z04.