#### Abstract

For with , the Schwab-Borchardt mean is defined as . In this paper, we find the greatest values of and and the least values of and in such that and . Similarly, we also find the greatest values of and and the least values of and in such that and . Here, , , and are the harmonic, arithmetic, and contraharmonic means, respectively, and , , , and are four Neuman means derived from the Schwab-Borchardt mean.

#### 1. Introduction

For with , the Schwab-Borchardt mean is defined as It is well known that the mean is strictly increasing in both and , nonsymmetric, and homogeneous of degree 1 in its variables. Several 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, respectively.

The Schwab-Borchardt mean was firstly investigated in . In , the authors pointed out that the logarithmic mean, two Seiffert means, and the Neuman-Sándor mean are particular cases of the Schwab-Borchardt mean. Later, and its special cases have been the subject of intensive research. In particular, many inequalities for them can be found in the literature .

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

Recently, the second author of this paper reviewed two elegant papers [14, 15] by Neuman and found that the bivariate means , , , and , derived from the Schwab-Borchardt mean are very interesting. They are defined as follows:

We call the means , , , and , defined in (4) the Neuman means. Moreover, if we let , then explicit formulas for , , , and are in the following: where , , , and are defined implicitly as , , and , respectively. Clearly, , , , and .

Neuman [14, 15] presented several optimal bounds for , , , and . The bounding quantities are arithmetic convex, geometric convex, and harmonic convex combinations of their generating means. Besides, he also proved that for with .

For fixed with , and . Let Then, it is not difficult to verify that and are continuous and strictly increasing on and , respectively. Note that , . Therefore, it is natural to ask what are the greatest values of and and the least values of and in such that and ? And what are the greatest values of and and the least values of and in such that and ? The main purpose of this paper is to answer these questions. Our main results are in Theorems 1 and 2.

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

Theorem 2. Let . Then, the double inequality holds for all with if and only if and . Also the double inequality holds for all with if and only if and .

#### 2. Two Lemmas

In order to prove the desired theorems, we first give two lemmas.

Lemma 1 (see [16, Theorem 1.25]). For , let be continuous on , and be 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. (1) The function is strictly increasing from onto .
(2) The function is strictly increasing from onto .
(3) The function is strictly decreasing from onto .
(4) The function is strictly decreasing from onto .

Proof. From part (1), let and . Then, , , and It is well known that is strictly decreasing on . Then, Lemma 1 and (14) lead to the conclusion that is strictly increasing on . Moreover, by l’Hôptial’s rule we have and .
From part (2), similarly let and . Then , and It is well known that is strictly increasing on . Then, by Lemma 1 and (15) we know that is strictly increasing on . Clearly, , while by l’Hôptial’s rule we have .
Parts (3) and (4) have been proven in [14, Theorem 3].

#### 3. Proofs of Theorems 1 and 2

Proof of Theorem 1. Without loss of generality, we assume that . Let and ; then, provided that . Thus, inequality (9) follows from (16) and Lemma 2(1). Similarly, provided that . Thus, inequality (10) follows from (17) and Lemma 2(2).

Proof of Theorem 2. Without loss of generality, we assume that . Let and , then provided that . Thus, inequality (11) follows from (18) and Lemma 2(3). Similarly, provided that . Thus, inequality (12) follows from (19) and Lemma 2(4).

#### Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 61374086 and 11171307, the Natural Science Foundation of Zhejiang Province under Grant LY13A010004, and the Natural Science Foundation of Huzhou Teachers College under Grant KX21063.