﻿ An Optimal Double Inequality among the One-Parameter, Arithmetic and Geometric Means

Journal of Applied Mathematics and Physics
Vol.1 No.7(2013), Article ID:40715,4 pages DOI:10.4236/jamp.2013.17001

An Optimal Double Inequality among the One-Parameter, Arithmetic and Geometric Means

Hongya Gao, Shuangli Li, Yanjie Zhang, Hong Tian

College of Mathematics and Computer Science, Hebei University, Baoding, China

Email: ghy@hbu.cn, 563211828@qq.com, 347764565@qq.com, 602580999@qq.com

Copyright © 2013 Hongya Gao 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.

Received October 12, 2013; revised November 12, 2013; accepted November 17, 2013

Keywords: Optimal Double Inequality; One-Parameter Mean; Arithmetic Mean; Geometric Mean

ABSTRACT

In the present paper, we answer the question: for fixed, what are the greatest value and the least value such that the double inequality holds for all with? where for, the one-parameter mean, arithmetic mean and geometric mean of two positive real numbers and are defined by and, respectively.

1. Introduction

For, the one-parameter mean, arithmetic mean and geometric mean of two positive real numbers and are defined by

(1)

and, respectively.

It is well-known that the one-parameter mean is continuous and strictly increasing with respect to for fixed with. Many means are special cases of the one-parameter mean, for example:

is the arithmetic mean, is the Heronian mean, is the geometric mean, and is the harmonic mean.

The one-parameter mean and its inequalities have been studied intensively, see [1-6].

The purpose of this paper is to answer the question: for, what are the greatest value and the least value such that the double inequality

holds for all with?

2. Main Result

The main result of this paper is the following theorem.

Theorem 2.1. Let. Then for any with, we have 1)

for2)

for3)

for.

The numbers and in 2) and 3) are optimal.

In order to prove Theorem 2.1, we need a preliminary lemma.

Lemma 2.1. For, one has

(2)

(3)

(4)

(5)

(6)

(2) follows from (3)-(6).

Proof of Theorem 2.1. Without loss of generality we assume and take We first consider the case. 1) follows from

From now on we assume Let then (1) leads to

(7)

where

(8)

where

(9)

(10)

(11)

where

(12)

(13)

where

(14)

(15)

(16)

We shall distinguish between two cases.

Case 1.. The left-hand side inequality of 2)

for follows from Lemma 2.1 because in this case

for all. In the sequel we assume.

We clearly see from (16) that

Thus is strictly decreasing for and strictly increasing for. (2.14) yields then for and for. The same reasoning applies to and as well, and noticing (13) and (12), one has

This result together with (11) implies

Thus is strictly decreasing for

and strictly increasing for The same reasoning applies to

and as well, and applying (8)-(10), we derive

Since for and for, then we know from (7) that

This implies the left-hand side of 2) and the right-hand side of 3).

Case 2.. From (14) we know that

From (13) we know that for and for. This implies is strictly decreasing for and strictly increasing for. From (12) we know

Therefore

(11) implies has the same property asthus is strictly decreasing for and strictly increasing for. The same reasoning applies to, and as well, and noticing (9) and (8), one has

which together with (7) implies

This implies the right-hand side of 2) and the left-hand side of 3).

We are now in the position to prove the constants

and are optimal.

For any (positive or negative, with sufficiently small) we consider the case. (12)

implies

By the continuity of, there exists such that

By (11), as the same property as. The same reasoning applies to, , and as well, and noticing (10)-(8), we know has the same property as. By (7) one has

This proves the optimality for.

To prove the optimality for in the right-hand side of 2) and the left-hand side of 3), we notice from

that there exists such that

for and and

for This ends the proof of Theorem 2.1.

3. Acknowledgements

This paper is supported by NSF of Hebei Province (A2011201011).

REFERENCES

1. H. Y. Gao and W. J. Niu, “Sharp Inequalities Related to One-Parameter Mean and Gini Mean,” Journal of Mathematical Inequalities, Vol. 6, No. 4, 2012, pp. 545-555.
2. W. S. Cheung and F. Qi, “Logarithmic Convexity of the One-Parameter Mean Values,” Taiwanese Journal of Mathematics, Vol. 11, No. 1, 2007, pp. 231-237.
3. M. K. Wang, Y. F. Qiu and Y. M. Chu, “An Optimal Double Inequality among the One-Parameter, Arithmetic and Harmonic Means,” Revue D’Analyse Numerique de Theorie de L’approximation, Vol. 39, No. 2, 2012, pp. 169-175.
4. H. N. Hu, G. Y. Tu and Y. M. Chu, “Optimal Bouds for the Seiffert Mean in Terms of One-Parameter Means,” Journal of Applied Mathematics, Vol. 2012, No. 1, 2012, Article ID: 917120.
5. B. Y. Long and Y. M. Chu, “Optimal Inequalities for Generalized Logarithmic, Arithmetic and Geometric Mean,” Journal of Inequalities and Applications, Vol. 2010, No. 1, 2010, Article ID: 806825.
6. N. G. Zheng, Z. H. Zhang and X. M. Zhang, Schur-Convexity of Two Types of One-Parameter Mean Values in Variables,” Journal of Inequalities and Applications, Vol. 2007, No. 1, 2007, Article ID: 78175.