- Research Article
- Open Access

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

- Ladislav Matejíčka
^{1}Email author

**2010**:902432

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

© Ladislav Matejíčka. 2010

**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.

## Keywords

- Real Number
- Unique Solution
- Open Problem
- Simple Computation
- Elementary Calculation

## 1. Introduction

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

Open Problem 1.

holds for and all with

Open Problem 2.

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 [1–19] and related references there in.

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

## 2. Main Result

## 3. Proof of Theorem 2.1

is nondecreasing.

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.

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

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## Copyright

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.