- Research Article
- Open Access

# Generalized Lazarevic's Inequality and Its Applications—Part II

- Ling Zhu
^{1}Email author

**2009**:379142

https://doi.org/10.1155/2009/379142

© Ling Zhu. 2009

**Received:**21 July 2009**Accepted:**30 November 2009**Published:**24 December 2009

## 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

## 1. Introduction

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

Theorem 1.1.

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.

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

Theorem 1.3.

## 2. Three Lemmas

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

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

Lemma 2.3.

Let and . Then the function strictly increases as increases.

## 3. A Concise Proof of Theorem 1.3

We compute

where

- (a)
Let , then and . Let for we have that and is decreasing for so is decreasing for and is decreasing on by Lemma 2.2. Hence is decreasing on and is decreasing on by Lemma 2.1. Thus is decreasing on by Lemma 2.1.

- (b)
Let , then . Let for we have that and is decreasing for so is decreasing for and is decreasing on by Lemma 2.2. Hence is increasing on and is decreasing on by Lemma 2.1. Thus is decreasing on by Lemma 2.1.

Since

the proof of Theorem 1.3 is complete.

## 4. Some New Lower Bounds for Logarithmic Mean

Assuming that and are two different positive numbers, let , , and be the arithmetic, geometric, and logarithmic means, respectively. It is well known that (see [2, 12–16])

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

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

Theorem 4.1.

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

## Authors’ Affiliations

## References

- Lazarević I:
**Neke nejednakosti sa hiperbolickim funkcijama.***Univerzitet u Beogradu. Publikacije Elektrotehničkog Fakulteta. Serija Matematika i Fizika*1966,**170:**41–48.Google Scholar - Mitrinović DS:
*Analytic Inequalities*. Springer, New York, NY, USA; 1970:xii+400.View ArticleMATHGoogle Scholar - Zhu L:
**On Wilker-type inequalities.***Mathematical Inequalities & Applications*2007,**10**(4):727–731.MathSciNetView ArticleMATHGoogle Scholar - Zhu L: Generalized Lazarevic's inequality and its applications—part I. submitted submittedGoogle Scholar
- Vamanamurthy MK, Vuorinen M:
**Inequalities for means.***Journal of Mathematical Analysis and Applications*1994,**183**(1):155–166. 10.1006/jmaa.1994.1137MathSciNetView ArticleMATHGoogle Scholar - Anderson GD, Qiu S-L, Vamanamurthy MK, Vuorinen M:
**Generalized elliptic integrals and modular equations.***Pacific Journal of Mathematics*2000,**192**(1):1–37. 10.2140/pjm.2000.192.1MathSciNetView ArticleGoogle Scholar - Pinelis I:
**L'Hospital type results for monotonicity, with applications.***Journal of Inequalities in Pure and Applied Mathematics*2002,**3**(1, article 5):1–5.MathSciNetMATHGoogle Scholar - Pinelis I:
**On L'Hospital-type rules for monotonicity.***Journal of Inequalities in Pure and Applied Mathematics*2006,**7**(2, article 40):1–19.MathSciNetMATHGoogle Scholar - Biernacki M, Krzyż J:
**On the monotonity of certain functionals in the theory of analytic functions.***Annales Universitatis Mariae Curie-Skłodowska*1955,**9:**135–147.MathSciNetMATHGoogle Scholar - Ponnusamy S, Vuorinen M:
**Asymptotic expansions and inequalities for hypergeometric functions.***Mathematika*1997,**44**(2):278–301. 10.1112/S0025579300012602MathSciNetView ArticleMATHGoogle Scholar - Alzer H, Qiu S-L:
**Monotonicity theorems and inequalities for the complete elliptic integrals.***Journal of Computational and Applied Mathematics*2004,**172**(2):289–312. 10.1016/j.cam.2004.02.009MathSciNetView ArticleMATHGoogle Scholar - Kuang JC:
*Applied Inequalities*. 3rd edition. Shangdong Science and Technology Press, Jinan City, China; 2004.Google Scholar - Sándor J:
**On the identric and logarithmic means.***Aequationes Mathematicae*1990,**40**(2–3):261–270.MathSciNetView ArticleMATHGoogle Scholar - Alzer H:
**Ungleichungen für Mittelwerte.***Archiv der Mathematik*1986,**47**(5):422–426. 10.1007/BF01189983MathSciNetView ArticleMATHGoogle Scholar - Stolarsky KB:
**The power and generalized logarithmic means.***The American Mathematical Monthly*1980,**87**(7):545–548. 10.2307/2321420MathSciNetView ArticleMATHGoogle Scholar - Zhu L:
**From chains for mean value inequalities to Mitrinovic's problem II.***International Journal of Mathematical Education in Science and Technology*2005,**36**(1):118–125. 10.1080/00207390412331314971View ArticleGoogle Scholar - Ostle B, Terwilliger HL:
**A comparison of two means.***Proceedings of the Montana Academy of Sciences*1957,**17:**69–70.Google Scholar - Leach EB, Sholander MC:
**Extended mean values. II.***Journal of Mathematical Analysis and Applications*1983,**92**(1):207–223. 10.1016/0022-247X(83)90280-9MathSciNetView ArticleMATHGoogle Scholar

## 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.