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Generalized Lazarevic's Inequality and Its Applications—Part II

  • Ling Zhu
Open Access
Research Article

Abstract

A generalized Lazarevic's inequality is established. The applications of this generalized Lazarevic's inequality give some new lower bounds for logarithmic mean.

Keywords

Lower Bound Concise Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Lazarević [1] (or see Mitrinović [2]) gives us the following result.

Theorem 1.1.

holds if and only if Open image in new window .

Recently, the author of this paper gives a new proof of the inequality (1.1) in [3] and extends the inequality (1.1) to the following result in [4].

Theorem 1.2.

holds if and only if Open image in new window .

Moreover, the inequality (1.1) can be extended as follows.

Theorem 1.3.

holds if and only if Open image in new window .

2. Three Lemmas

Lemma 2.1 (see [5, 6, 7, 8]).

Let Open image in new window be two continuous functions which are differentiable on Open image in new window . Further, let Open image in new window on Open image in new window . If Open image in new window is increasing (or decreasing) on Open image in new window , then the functions Open image in new window and Open image in new window are also increasing (or decreasing) on Open image in new window .

Lemma 2.2 (see [9, 10, 11]).

Let Open image in new window and Open image in new window be real numbers, and let the power series Open image in new window and Open image in new window be convergent for Open image in new window . If Open image in new window for Open image in new window and if Open image in new window is strictly increasing (or decreasing) for Open image in new window then the function Open image in new window is strictly increasing (or decreasing) on Open image in new window .

Lemma 2.3.

Let Open image in new window and Open image in new window . Then the function Open image in new window strictly increases as Open image in new window increases.

3. A Concise Proof of Theorem 1.3

Let Open image in new window , where Open image in new window , and Open image in new window . Then

where Open image in new window , and Open image in new window .

We compute

where

and Open image in new window .

Since

the proof of Theorem 1.3 is complete.

4. Some New Lower Bounds for Logarithmic Mean

Assuming that Open image in new window and Open image in new window are two different positive numbers, let Open image in new window , Open image in new window , and Open image in new window be the arithmetic, geometric, and logarithmic means, respectively. It is well known that (see [2, 12, 13, 14, 15, 16])

Ostle and Terwilliger [17] (or see Leach and Sholander [18], Zhu [16]) gave bounds for Open image in new window in terms of Open image in new window and Open image in new window as follows:

Without loss of generality, let Open image in new window and Open image in new window , then Open image in new window . Replacing Open image in new window with Open image in new window in (1.3), we obtain the following new results for three classical means.

Theorem 4.1.

holds if and only if Open image in new window .

Now letting Open image in new window in inequality (4.3) be Open image in new window , and Open image in new window , respectively, by Theorem 4.1 and Lemma 2.3 we have the following inequalities:

References

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Copyright information

© Ling Zhu. 2009

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.

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Gongshang UniversityHangzhouChina

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