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Chinese Journal of Mathematics
Volume 2013 (2013), Article ID 852516, 7 pages
http://dx.doi.org/10.1155/2013/852516
Research Article

Optimal Lehmer Mean Bounds for the Combinations of Identric and Logarithmic Means

1College of Nursing, Huzhou Teachers College, Huzhou 313000, China
2School of Mathematics and Computation Science, Hunan City University, Yiyang 413000, China

Received 11 July 2013; Accepted 5 August 2013

Academic Editors: M. Coppens, Y. Miao, and P.-y. Nie

Copyright © 2013 Xu-Hui Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For any , we answer the questions: what are the greatest values and and the least values and , such that the inequalities and hold for all with ? Here, , , and denote the identric, logarithmic, and th Lehmer means of two positive numbers and , respectively.

1. Introduction

For with , the logarithmic mean and identric mean are defined by respectively. In the recent past, both mean values have been the subject of intensive research. In particular, many remarkable inequalities for and can be found in the literature [119]. In [14, 17, 20], inequalities between , , and the classical arithmetic-geometric mean of Gauss are proved. The ratio of identric means leads to the weighted geometric mean which has been investigated in [11, 13, 21]. It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [2224]. In [22], the authors study a variant of Jensen’s functional equation involving , which appears in a heat conduction problem. A representation of as an infinite product and an iterative algorithm for computing the logarithmic mean as the common limit of two sequences of special geometric and arithmetic means are given in [5]. In [25, 26], it is shown that can be expressed in terms of Gauss’ hypergeometric function . And in [26], the authors prove that the reciprocal of the logarithmic mean is strictly totally positive; that is, every determinant with elements , where and , is positive for all .

For , the power mean and Lehmer mean of order of two positive numbers and with are defined by

It is well known that and are strictly increasing with respect to for fixed with . The main properties for and are given in [2732].

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

The following sharp bounds for , , , and in terms of power means are proved in [24, 6, 8, 9, 33]: for all with .

Alzer and Qiu [19] proved that for all with if and only if , , and .

The following sharp upper and lower Lehmer mean bounds for , , , and are presented in [34]: for all with .

The purpose of this paper is to present the best possible upper and lower Lehmer mean bounds of the product and the sum for any and all with .

2. Lemmas

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

Lemma 1. If , then the following statements are true:(1);(2);(3);(4);(5);(6).

Proof. (1) We clearly see that
(2) Let Then simple computations lead to
Therefore, Lemma 1(2) follows from (14) and (15).
(3) Let Then simple computations yield
Therefore, Lemma 1(3) follows from (16) and (17).
(4) Let Then simple computations lead to
Therefore, Lemma 1(4) follows from (18) and (19).
(5) Let Then simple computations yield
It follows from (28) and the discriminant of the quadratic function that
Therefore, Lemma 1(5) follows from (20)–(27) and (29).
(6) Let Then we have
Therefore, Lemma 1(6) follows from (30) and (31).

Lemma 2. Suppose that . If , then for .

Proof. Let , , , and /; then elaborated computations lead to
It follows from Lemma 1(6) and (49) that for .
From Lemma 1(1)–(5) and (38)–(48), we clearly see that
Therefore, Lemma 2 follows from and (32)–(36) together with (50) and (51).

Lemma 3. Inequality holds for any and all with .

Proof. Without loss of generality, we assume that . Let and ; then , , and from (1) and (4) we have Let Then simple computations lead to where is defined as in Lemma 2.
From (54) and (55) together with Lemma 2, we clearly see that for .
Therefore, Lemma 3 follows from (52) and (53) together with (56).

3. Main Results

Theorem 4. Inequality holds for any and all with , and and are the best possible lower and upper Lehmer mean bounds for the product .

Proof. Inequality follows directly from (10) and (11).
For the other inequality, we note that
Therefore, follows from (7) and (10) together with Lemma 3 and (57).
Next, we prove that and are the best possible lower and upper Lehmer mean bounds for the product .
For any and , from (1) and (2) we have Letting and making use of Taylor expansion, one has
Equations (58) and (59) imply that for any , there exist and , such that for and for .

Theorem 5. Inequality holds for any and all with , and and are the best possible lower and upper Lehmer mean bounds for the sum .

Proof. Inequality follows directly from (10) and (11), and inequality follows from Theorem 4.
Next, we prove that and are the best possible lower and upper Lehmer mean bounds for the sum .
For any and , from (1) and (2), we have Letting and making use of Taylor expansion, one has
Equations (60)-(62) imply that for any , there exist and , such that for and for .

Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 11071069 and 11171307 and the Natural Science Foundation of Zhejiang Province under Grants LY13H070004 and LY13A010004.

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