Sharp Power Mean Bounds for Sándor Mean

Abstract

We prove that the double inequality holds for all with if and only if and …, where and are the Sándor and th power means of and , respectively.

1. Introduction

Let and with . Then the th power mean of and is given by

The main properties for the power mean are given in [1]. It is well known that is strictly increasing with respect to for fixed with . Many classical means are the special cases of the power mean; for example, is the harmonic mean, is the geometric mean, is the arithmetic mean, and is the quadratic mean.

Let , , , , and be the logarithmic, first Seiffert, identric, Neuman-Sándor, and second Seiffert means of two distinct positive real numbers and , respectively. Then it is well known that the inequalities hold for all with .

Recently, the bounds for certain bivariate means in terms of the power mean have been the subject of intensive research. Seiffert [2] proved that the inequalities hold for all with .

Jagers [3] proved that the double inequality holds for all with .

In [4, 5], Hästö established that for all with .

Witkowski [6] proved that the double inequality holds for all with .

In [7], Costin and Toader presented that for all with .

Chu and Long [8] proved that the double inequality holds for all with if and only if and .

The following sharp bounds for the logarithmic and identric means in terms of the power means can be found in the literature [9–16]: for all with .

Recently, Sándor [17] introduced the Sándor mean of two positive real numbers and , which is given by

In [18], Sándor proved that for all with .

In the Introduction we cite only a minor part of the existing literature on the considered means. For example, an important paper on the first Seiffert mean is again due to Sándor [19].

The main purpose of this paper is to present the best possible parameters and such that the double inequality holds for all with .

2. Lemmas

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

Lemma 1. Let be defined by Then(1) is strictly decreasing with respect to on if and only if ;(2) is strictly increasing with respect to on if and only if .

Proof. It follows from (12) that where
(1) If is strictly decreasing with respect to on , then (13) leads to the conclusion that for all . In particular, we have . We assert that . Indeed, from (14) we clearly see that , if , and if .
If , then it follows from (14) that for all .
Equation (14) and inequality (15) lead to the conclusion that for all .
Therefore, is strictly decreasing with respect to on which follows from (13) and (16).
(2) If is strictly increasing with respect to on , then (13) leads to the conclusion that for all . In particular, we have and .
If , then (14) and (15) lead to the conclusion that for all .
Therefore, is strictly increasing with respect to on which follows from (13) and (18).

Lemma 2. Let be defined by (12). Then there exists such that is strictly increasing with respect to on and strictly decreasing with respect to on if .

Proof. Let and be defined by (14). Then (14) leads to for .
Inequality (24) implies that is strictly convex with respect to on . From (22) and (23) together with the strict convexity of with respect to on we clearly see that there exists such that is strictly decreasing with respect to on and strictly increasing with respect to on . We assert that
Indeed, if , then it follows from (20) and the piecewise monotonicity of with respect to on that is strictly increasing with respect to on . Hence, we get for all . This conjunction with Lemma 1 and (13) leads to the conclusion that , which contradicts with .
From (20) and (21) together with (25) and the piecewise monotonicity of with respect to on we clearly see that there exist and such that is strictly increasing with respect to on and strictly decreasing with respect to on .
Therefore, Lemma 2 follows easily from (13) and (19) together with the piecewise monotonicity of with respect to on .

Lemma 3. Let be defined by (12). Then the following statements are true:(1) for all if and only if ;(2) for all if and only if ;(3)if , then there exists such that , for , and for .

Proof. (1) If for all , then . Therefore, follows from for .
If , then Lemma 1 (1) leads to the conclusion that for all .
(2) If for all , then by making use of L’Höspital’s rules and (12) we get and .
If , then Lemma 1 (2) leads to the conclusion that for all .
(3) If , then it follows from (12) that
Therefore, Lemma 3 (3) follows from Lemma 2 and (27).

Lemma 4. Let be defined by Then(1) is strictly increasing with respect to on if and only if ;(2) is strictly decreasing with respect to on if and only if ;(3)if , there exists such that is strictly decreasing with respect to on and strictly increasing with respect to on .

Proof. It follows from (28) that where is defined by (12).
Therefore, Lemma 4 follows from Lemma 3 and (29).

3. Main Results

Theorem 5. The double inequality holds for all with if and only if and .

Proof. Since both the Sándor mean and th power mean are symmetric and homogeneous of degree 1, without loss of generality, we assume that and .
We first prove that the inequality holds for all if and only if .
If , then from (28) and Lemma 4 (2) we get for all .
Therefore, for all and follows from (31) and the monotonicity of the function .
If , then (28) leads to for all . In particular, we have and .
Next, we prove that the inequality holds for all if and only if .
If holds for all , then (28) leads to for all . In particular, we have and .
If , then (28) leads to
It follows from (28) and (34) together with Lemma 4 (3) that for all .
Therefore, for all and follows from (35) and the monotonicity of the function .

Theorem 6. Let with . Then the double inequality holds with the best possible constants and .

Proof. Since both the Sándor mean and th power mean are symmetric and homogeneous of degree 1, without loss of generality, we assume that and . It follows from Lemma 4 (1) and (2) together with (28) that for all .
Therefore, for all follows from (37), and the optimality of the parameters and follows from the monotonicity of the functions and .

Remark 7. For all with . Then from Lemma 4 (1) and (2) together with (28) we clearly see that the Ky Fan type inequalities hold if and only if and .

Let and be the th Lehmer mean of two positive real numbers and . Then the function defined by (12) can be rewritten as

From Lemma 3 and (39) we get Remark 8 as follows.

Remark 8. The double inequality holds for all with if and only if and .

From (5) and (9) together with Theorem 5 one has the following.

Remark 9. The inequalities hold for all with .

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 11171307 and 61374086 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.