- Research Article
- Open Access
Proof of One Optimal Inequality for Generalized Logarithmic, Arithmetic, and Geometric Means
© Ladislav Matejíčka. 2010
- Received: 11 July 2010
- Accepted: 31 October 2010
- Published: 2 November 2010
Two open problems were posed in the work of Long and Chu (2010). In this paper, we give the solutions of these problems.
- Real Number
- Unique Solution
- Open Problem
- Simple Computation
- Elementary Calculation
In the paper , 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.
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.
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.
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