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.
In the paper , Long and Chu propose the two following open problems:
Open Problem 1.
Open Problem 2.
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
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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