Open Access

Proof of One Optimal Inequality for Generalized Logarithmic, Arithmetic, and Geometric Means

Journal of Inequalities and Applications20102010:902432

https://doi.org/10.1155/2010/902432

Received: 11 July 2010

Accepted: 31 October 2010

Published: 2 November 2010

Abstract

Two open problems were posed in the work of Long and Chu (2010). In this paper, we give the solutions of these problems.

1. Introduction

The arithmetic and geometric means of two positive numbers and are defined by , , respectively. If is a real number, then the generalized logarithmic mean with parameter of two positive numbers , is defined by
(1.1)

In the paper [1], Long and Chu propose the two following open problems:

Open Problem 1.

What is the least value such that the inequality
(1.2)

holds for and all with

Open Problem 2.

What is the greatest value such that the inequality
(1.3)

holds for and all with

For information on the history, background, properties, and applications of inequalities for generalized logarithmic, arithmetic, and geometric means, please refer to [119] and related references there in.

The aim of this article is to prove the following Theorem 2.1.

2. Main Result

Theorem 2.1.

Let , , , . Let be a solution of
(2.1)
Then,
(2.2)
and is the best constant,
(2.3)

and is the best constant.

3. Proof of Theorem 2.1

Because is increasing with respect to for fixed and , it suffices to prove that for any (resp., ) there exists such that (resp., ), and is the best constant. Without loss of generality, we assume that . Let , . Equations (2.2), (2.3) are equivalent to
(3.1)
On putting , we obtain (3.1) is equivalent to
(3.2)
Introduce the function by
(3.3)
Simple computations yield for
(3.4)
Let and the unique solution to
(3.5)
To see that is optimal in both cases (2.2), (2.3), note that . Thus, if the constant is decreased (resp., increased), then the desired bound for would not hold for small . This follows from the fact that for a fixed , the function
(3.6)

is nondecreasing.

From now on, let for . To show the estimates for this , we start from observing that . Furthermore, one easily checks that
(3.7)
Thus, it suffices to verify that has exactly one zero inside the interval . It follows from the mean value theorem. After some computations, this is equivalent to saying that the function given by
(3.8)

has exactly one root in . Here, the expression under the logarithm may be nonpositive, so we define on a maximal interval, contained in . It is easy to see that this interval must be of the form , for some . This follows from the fact that is strictly positive on and is strictly increasing on this interval.

Since and , we will be done if we show that has exactly one root in . After some computations, we obtain that the equation is equivalent to
(3.9)
Because is a quadratic polynomial in the variable , all that remains is to show that
(3.10)
or, in virtue of the definition of ,
(3.11)

This can be easily established by some elementary calculations. It completes the proof.

Declarations

Acknowledgments

The author is indebted to the anonymous referee for many valuable comments, for a correction of one part of the proof, and for his improving of the organization of the paper. This work was supported by Vega no. 1/0157/08 and Kega no. 3/7414/09.

Authors’ Affiliations

(1)
Faculty of Industrial Technologies in Púchov, Alexander Dubček University in Trenčín

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Copyright

© Ladislav Matejíčka. 2010

This article is published under license to BioMed Central Ltd. 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.