# The Best Lower Bound Depended on Two Fixed Variables for Jensen's Inequality with Ordered Variables

- Vasile Cirtoaje
^{1}Email author

**2010**:128258

https://doi.org/10.1155/2010/128258

© Vasile Cirtoaje. 2010

**Received: **16 June 2010

**Accepted: **4 November 2010

**Published: **24 November 2010

## Abstract

We give the best lower bound for the weighted Jensen's discrete inequality with ordered variables applied to a convex function , in the case when the lower bound depends on , weights, and two given variables. Furthermore, under the same conditions, we give some sharp lower bounds for the weighted AM-GM inequality and AM-HM inequality.

## 1. Introduction

## 2. Main Results

Theorem 2.1.

For proving Theorem 2.1, we will need the following three lemmas.

Lemma 2.2.

Lemma 2.3.

Lemma 2.4.

Applying Theorem 2.1 for and using the substitutions , , we obtain

Corollary 2.5.

Using Corollary 2.5, we can prove the propositions below.

Proposition 2.6.

with equality for . When , equality holds again for , , .

Proposition 2.7.

with equality if and only if .

Remark 2.8.

Equality in (2.18) holds for . If , then equality holds again for , , .

Remark 2.9.

with equality if and only if .

Applying Theorem 2.1 for , we obtain

Corollary 2.10.

Remark 2.11.

Applying Theorem 2.1 for , we obtain the following.

Corollary 2.12.

Using Corollary 2.12, we can prove the following proposition.

Proposition 2.13.

## 3. Proof of Lemmas

Proof of Lemma 2.2.

Proof of Lemma 2.3.

by Lemma 2.2, the conclusion follows.

Proof of Lemma 2.4.

## 4. Proof of Theorem

## 5. Proof of Propositions

Proof of Proposition 2.6.

Since for , and is increasing, , is increasing, and hence for . This concludes the proof.

Proof of Proposition 2.7.

Since for , is strictly increasing, , and is strictly increasing, , is strictly increasing, and hence for .

Proof of Proposition 2.13.

The proposition is proved.

## Authors’ Affiliations

## References

- Mitrinović DS, Pečarić JE, Fink AM:
*Classical and New Inequalities in Analysis, Mathematics and Its Applications*.*Volume 61*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740.View ArticleMATHGoogle Scholar - Dragomir SS, Pečarić J, Persson LE: Properties of some functionals related to Jensen's inequality.
*Acta Mathematica Hungarica*1996, 70(1–2):129–143. 10.1007/BF00113918MathSciNetView 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.