Optimal Lower Generalized Logarithmic Mean Bound for the Seiffert Mean

Abstract

We present the greatest value such that the inequality holds for all with , where and denote the Seiffert and th generalized logarithmic means of and , respectively.

1. Introduction

For , the th generalized logarithmic mean is defined by and the Seiffert mean [1] is defined by

It is well known that the generalized logarithmic mean is continuous and strictly increasing with respect to for fixed with . The special cases of the generalized logarithmic mean are, for example, is the geometric mean,   is the logarithmic mean, is the identric mean, and is the arithmetic mean. The Seiffert mean can be rewritten as (see [2, equation (2.4)])

Recently, the bivariate means have been the subject of intensive research. In particular, many remarkable inequalities and properties for the generalized logarithmic and the Seiffert means can be found in the literature [3–13].

In [1, 11], Seiffert proved that the inequalities hold for all with .

Sándor [14] presented the bounds for the Seiffert mean in terms of the arithmetic mean and geometric mean as follows: for all with .

Hästö [15] proved that the double inequality holds for all with if and only if and , where is the th power mean of and .

In [16], the authors found the greatest value and least value such that the double inequality holds for all with , where is the contraharmonic mean of and .

Motivated by the first inequality in (4), Gao [17] gave the best possible constants and such that the double inequality holds for all with .

In [18], the author solved the following open problem proposed by Long and Chu [19]: what is the smallest (largest ) such that the inequality holds for and all with ?

Chu et al. [20] proved that the double inequality holds for all with if and only if and , where is the second Seiffert mean of and , and is the unique solution of the equation .

In [21], the authors answered the question: what are the greatest value and the least value such that the double inequality holds for any with and all with ?

Motivated by the first inequality in (4), it is natural to ask what are the best possible generalized logarithmic mean bounds for the Seiffert mean ? It is the aim of this paper to answer this question.

2. Preliminaries

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

Lemma 1. Let the function be defined with Then, is a continuous and strictly increasing function.

Proof. From (6), we clearly see that If , then simple computation yields If we define then Equation (11) implies that Equations (10) and (12) lead to for .
From (9) and (10) together with (13), we clearly see that for .
Therefore, the continuity of follows from (6) and (7), and the strict monotonicity of follows from (8), (14) and the continuity of .

Remark 2. From Lemma 1, we clearly see that for any fixed , there exists a unique such . In particular, for , making use of Mathematica software, we get
Therefore, the unique solution of the equation belongs to the interval .

Lemma 3. Let and let be defined with    . Then, there exists such that for and for .

Proof. Simple computations lead to
Let . Further computations lead to
Inequalities (34) and (35) imply that for .
From (31) and (32) together with (36), we clearly see that there exists such that for and for . Hence, is strictly decreasing on and strictly increasing on .
It follows from (28) and (29) together with the monotonicity of that there exists such that is strictly decreasing on and strictly increasing on .
Equations (25)-(26) and the monotonicity of lead to the conclusion that there exists such that for and for . Therefore, is strictly decreasing on and strictly increasing on .
From (22) and (23) together with the monotonicity of , we clearly see that there exists such that is strictly decreasing on and strictly increasing on .
Equations (19)-(20) and the monotonicity of imply that there exists such that is strictly decreasing on and strictly increasing on .
Therefore, Lemma 3 follows from (16) and (17) together with the monotonicity of .

3. Main Results

Theorem 4. Let be as in Lemma 1 and let be the unique solution of the equation . Then, for all , , the inequality holds, and is the best possible lower generalized logarithmic mean bound for the Seiffert mean .

Proof. From Remark 2, it follows that
We first prove that the inequality (37) holds. Without loss of generality, we assume that . If , then, from (1) and (2), we have If then simple computations lead to where where
From (38) and (44)-(45) together with Lemma 3, we clearly see that there exists such that is strictly decreasing on and strictly increasing on . Then, (38) and (42)-(43) together with the monotonicity of imply that there exists such that is strictly decreasing on and strictly increasing on .
Therefore, follows from (39)–(41) and the monotonicity of .
Next, we prove that is the best possible lower generalized logarithmic mean bound for the Seiffert mean .
For any , from (1) and (2), we get Lemma 1 and (46) lead to
Inequality (47) implies that, for , there exists such that for .