Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 914242 | https://doi.org/10.1155/2014/914242

Zhi-Jun Guo, Yu-Ming Chu, Ying-Qing Song, Xiao-Jing Tao, "Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means", Abstract and Applied Analysis, vol. 2014, Article ID 914242, 8 pages, 2014. https://doi.org/10.1155/2014/914242

Sharp Bounds for Neuman Means by Harmonic, Arithmetic, and Contraharmonic Means

Academic Editor: Shuangjie Peng
Received20 May 2014
Accepted12 Jul 2014
Published23 Jul 2014

Abstract

We give several sharp bounds for the Neuman means and ( and ) in terms of harmonic mean H (contraharmonic mean C) or the geometric convex combination of arithmetic mean A and harmonic mean H (contraharmonic mean C and arithmetic mean A) and present a new chain of inequalities for certain bivariate means.

1. Introduction

For with , the Schwab-Borchardt mean [13] of and is defined as where and are the inverse cosine and inverse hyperbolic cosine functions, respectively.

The Schwab-Borchardt mean can be expressed by the symmetric elliptic integral [4] of the first kind as follows [5] (see also [6, (3.21)]): where .

Recently, the Schwab-Borchardt mean has been the subject of intensive research. In particular, many remarkable inequalities for the Schwab-Borchardt mean and its generated means can be found in the literature [13, 710].

Very recently, Neuman [11] found a new mean derived from the Schwab-Borchardt mean as follows:

Let , , , , , , , , and be, respectively, the geometric, harmonic, logarithmic, first Seiffert, arithmetic, Neuman-Sándor, second Seiffert, quadratic, and contraharmonic means, and let be the Neuman means. Then Neuman [11] proved that for all with , and the double inequalities hold for all with if and only if , , , , , , , and .

Zhang et al. [12] presented the best possible parameters and such that the double inequalities hold for all with .

In [13], the authors found the greatest values , , , , , , , and and the least values , , , , , , , and such that the double inequalities hold for all with .

Let , , , , , and be the parameters such that , . Then He et al. [14] proved that

Let , , , and . Then we clearly see that for all with , and the functions and are, respectively, strictly increasing on the intervals and for fixed with .

The main purpose of this paper is to find the best possible parameters , , , , , , , , , and such that the double inequalities hold, for all with , and present a new chain of inequalities for certain bivariate means.

2. Lemmas

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

Lemma 1 (see [15, Theorem 1.25]). For , let be continuous on and differentiable on ; let on . If is increasing (decreasing) on , then so are If is strictly monotone, then the monotonicity in the conclusion is also strict.

Lemma 2 (see [16, Lemma 1.1]). Suppose that the power series and have the radius of convergence and for all . If the sequence is (strictly) increasing (decreasing) for all , then the function is also (strictly) increasing (decreasing) on .

Lemma 3. The function is strictly decreasing from onto .

Proof. Making use of power series expansion we get
Let Then and is strictly decreasing for all .
Note that
Therefore, Lemma 3 follows easily from Lemma 2 and (18)–(21) together with the monotonicity of the sequence .

Lemma 4. The function is strictly increasing from onto .

Proof. Let and . Then simple computations lead to and is strictly increasing on .
Note that
Therefore, Lemma 4 follows from Lemma 1, (23), (24), and the monotonicity of .

Lemma 5. The function is strictly decreasing from onto .

Proof. Let and . Then simple computations lead to
Let Then it is not difficult to verify that for all , where and .
Note that
It follows from Lemma 2 and (27)–(29) that the function is strictly decreasing on . Therefore, Lemma 5 follows from (26) and (30) together with Lemma 1 and the monotonicity of .

Lemma 6. The function is strictly decreasing from onto .

Proof. Let , , , and . Then simple computations lead to
Since the function is strictly decreasing on , hence (34) leads to the conclusion that is strictly decreasing on .
Note that
Therefore, Lemma 6 follows easily from (32), (33), (35), and Lemma 1 together with the monotonicity of .

Lemma 7. The function is strictly increasing from onto .

Proof. Let and . Then simple computations lead to
Let
Then for all .
It follows from Lemma 2 and (38)–(40) that is strictly increasing on .
Note that
Therefore, Lemma 7 follows from Lemma 1, (37), and (41) together with the monotonicity of .

Lemma 8. The function is strictly increasing from onto .

Proof. Let , , , and . Then simple computations lead to
Since the function is strictly decreasing on , hence Lemma 1 and (43) lead to the conclusion that is strictly increasing on .
Note that
Therefore, Lemma 8 follows easily from (44) and (45) together with the monotonicity of .

3. Main Results

Theorem 9. The double inequalities hold for all with if and only if , , , , , , , and .

Proof. Without loss of generality, we assume that . Let , , , , and be the parameters such that , . Then (9)–(12) lead to the conclusion that inequalities (46)–(49) are, respectively, equivalent to
Therefore, Theorem 9 follows easily from (50)–(53) and Lemmas 58.

Theorem 10. Let . Then the double inequalities, hold for all with if and only if , , , and .

Proof. Without loss of generality, we assume that . Let , , , and be the parameters such that . Then from (9) and (10) we have
Therefore, Theorem 10 follows easily from (55) and (56) together with Lemmas 3 and 4.

Theorem 11. Let . Then the double inequalities, hold for all with if and only if , , , and .

Proof. Without loss of generality, we assume that . Let , , , and be the parameters such that . Then from (11) and (12) one has where the functions and are defined as in Lemmas 3 and 4, respectively.
Note that
Therefore, Theorem 11 follows easily from Lemmas 3 and 4 together with (58)-(59).

Theorem 12. Let , , , and . Then the inequalities