Best Possible Bounds for Neuman-Sándor Mean by the Identric, Quadratic and Contraharmonic Means

Abstract

We prove that the double inequalities hold for all with if and only if , , , and , where , , , and are the identric, Neuman-Sándor, quadratic, and contraharmonic means of and , respectively.

1. Introduction

For and with , the identric mean , Neuman-Sándor mean [1], quadratic mean , contraharmonic mean , and th power mean are defined by respectively, where is the inverse hyperbolic sine function.

Recently, the identric, Neuman-Sándor, quadratic, and contraharmonic means have attracted the interest of numerous eminent mathematicians. In particular, many remarkable inequalities for these means can be found in the literature [1–18].

Let , , , , , and be the harmonic, geometric, logarithmic, first Seiffert, arithmetic, and second Seiffert means of two distinct positive numbers and , respectively. Then it is well known that the inequalities hold for all with .

Neuman and Sándor [1, 8] established that for all with .

Let with , , and . Then the Ky Fan inequalities were presented in [1].

Li et al. [19] found the best possible bounds for the Neuman-Sándor mean in terms of the generalized logarithmic mean . Neuman [20] and Zhao et al. [21] proved that the inequalities hold for all with if and only if , , , , , , , and .

In [22], Chu and Long gave the best possible constants , and such that the double inequalities and hold for all with .

The ratio of identric means leads to the weighted geometric mean which has been investigated in [23–25]. Alzer [26] proved that the inequalities hold for all with .

The following sharp bounds for , , and in terms of the power mean and the convex combination of arithmetic and geometric means are given in [27] as for all with .

Chu et al. [28] presented the optimal constants , and such that the double inequalities hold for all with .

The aim of this paper is to find the best possible constants and such that the double inequalities hold for all with . All numerical computations are carried out using MATHEMATICA software.

2. Lemmas

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

Lemma 1. The double inequality holds for .

Proof. To prove Lemma 1, it suffices to prove that for .
From the expressions of and , we get where for .
Therefore, inequality (12) follows from (14)–(16), and inequality (13) follows from (14)–(17).

Lemma 2. Let Then for , and for .

Proof. To prove inequalities (19) and (20), it suffices to show that for , and for .
From (21) and (22), one has for , and for .
Therefore, inequality (21) follows from (23) and (24), and inequality (22) follows from (23) and (25).

Lemma 3. Let
Then the double inequality holds for .

Proof. To prove inequality (27), it suffices to show that for .
First, we prove inequality (28). From the expression of , we have where
Equation (32) leads to where
Note that for , and for .
It follows from (32) and (34)–(36) together with Lemma 1 that for , and for .
From (33), (37), and (38), we clearly see that there exists such that for and for . Then (31) leads to the conclusion that is strictly increasing on and strictly decreasing on .
Therefore, inequality (28) follows from (30) and the piecewise monotonicity of .
Next, we prove inequality (29). From the expression of , we get where
It follows from Lemma 1 and (40) that for .
Therefore, inequality (29) follows from (39) together with (41).