- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

The Scientific World Journal

Volume 2013 (2013), Article ID 645201, 4 pages

http://dx.doi.org/10.1155/2013/645201

## On Satnoianu-Wu’s Inequality

College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia 028043, China

Received 22 April 2013; Accepted 28 June 2013

Academic Editors: F. J. Garcia-Pacheco, Y. Sawano, and S. Tikhonov

Copyright © 2013 Bo-Yan Xi. 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.

#### Abstract

By applying techniques in the theory of convex functions and Schur-geometrically convex functions, the author investigates a conjecture of Satnoianu on an algebraic inequality and generalizes some known results in recent years.

#### 1. Introduction

For and with , let These quantities are, respectively, called the arithmetic, geometric, and harmonic means of a positive sequence . For more information on the theory of means, see the monograph [1] or the papers [2–5] and plenty of references therein.

For convenience, in what follows, we use the notation for and .

In [6], Satnoianu posed the following conjecture.

Conjecture 1. *For , , and , it is valid that
*

This conjecture has been solved and researched in [7–10]. Among them, Wu obtained in [10] the following result.

Theorem 2 (see [10, Theorem 1]). *Let be a positive integer and let , , and for be positive numbers. If and
**
then
*

It is clear that inequality (5) generalizes (3). Therefore, we would like to call inequality (5) Satnoianu-Wu’s inequality.

Now we naturally pose the following questions: (1)Can one improve the condition (4)? (2)Does the reversed inequality of (5) exist? (3)Are there other types of inequalities of Satnoianu-Wu type?

The goal of this paper is to answer these questions.

#### 2. Definitions and Lemmas

We need the following definitions and lemmas.

*Definition 3 (see [11, page 8]). *Let and . We say that is majorized by (in symbols ) if
for , where and are rearrangements of and in a descending order.

*Definition 4 (see [12]). *Let .(1)The set is said to be geometrically convex if for every and .(2)A function is said to be Schur-geometrically convex on if implies for every .(3)A function is said to be Schur-geometrically concave on if implies for every .

Lemma 5 (see [12]). *Let be a symmetric and geometrically convex set with inner points and a symmetric and differentiable function in . Then is a Schur-geometrically convex (or Schur-geometrically concave, resp.) function on if and only if
*

Lemma 6 (see [1, page 4, Bernoulli's inequality]). *The inequality
**
holds for and or for and . If and , inequality (8) is reversed.*

#### 3. Main Results

Now we start off to demonstrate our main results.

Theorem 7. *Let be defined as in (2) and , let for , and let with . If , then
**
If , inequality (9) is reversed.*

*Proof. *For and with , since
then(1)for , or for and , the function is convex on ;(2)for , or for and , the function is concave on .By Jensen’s inequality, when , or when and , we have
where . When , or when and , inequality (11) is reversed.

Let for , , , and . Then it is easy to show that . Making use of (11), we obtain inequality (9). The proof of Theorem 7 is complete.

Corollary 8. *Under the conditions of Theorem 7 and when , if , then
**
if , inequality (12) is reversed.*

Corollary 9. *Let , for , and . If , then
**
if , inequality (13) is reversed.*

*Proof. *This follows from Theorem 7 by considering that if , we have and that if , we have .

Theorem 10. *Let , for , with , and defined by (2). If , then
**
if , inequality (14) is reversed.*

*Proof. *For , let
Then
for . When , or when and , we have
when and , inequality (17) is reversed.

Using Lemma 5, we have the following conclusions:(1)if , or if and , the function is Schur-geometrically convex on ;(2)if and , the function is Schur-geometrically concave on .By the fact that
and by Definition 4, if , or if and , we have
if and , inequality (19) is reversed.

Letting for leads to . Putting , and in inequality (19) results in inequality (14). The proof of Theorem 10 is complete.

Corollary 11. *Under the conditions of Theorem 10 and when , if , then
**
if , inequality (20) is reversed.*

*Remark 12. *When and , from Lemma 6 and (4), it follows that

Theorem 13. *Let , for , and with . If , then
**
if , inequality (22) is reversed.*

*Proof. *Since is a convex (or concave, resp.) function on for or (or for , resp.), by Jensen's inequality, if or , we have
if , inequality (23) is reversed.

Letting and shows . Further from (23), we obtain inequality (22). The proof of Theorem 13 is complete.

*Remark 14. *It is clear that inequalities (9) and (22) both generalize inequality (3).

Corollary 15. *Let , for , with , and defined as in (2).*(1)*When , one has
*(2)*When and , one has
*(3)*When and , one has
*(4)*When and , one has
*

*Proof. *This follows from utilizing the well-known harmonic-geometric-arithmetic mean inequality
and Theorems 7 to 13.

Corollary 16. *Under the conditions of Corollary 15 and when ,*(1)*if and , one has
*(2)*if and , one has
*(3)*if and , one has
*

#### Acknowledgments

The author thanks four anonymous referees for their careful corrections to and valuable comments on the original version of this paper. This work was partially supported by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY13159, China.

#### References

- P. S. Bullen,
*Handbook of Means and Their Inequalities*, vol. 560 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. - B.-N. Guo and F. Qi, “A simple proof of logarithmic convexity of extended mean values,”
*Numerical Algorithms*, vol. 52, no. 1, pp. 89–92, 2009. View at Publisher · View at Google Scholar · View at Scopus - B.-N. Guo and F. Qi, “The function (b
^{x}−a^{x})/x: logarithmic convexity and applications to extended mean values,”*Filomat*, vol. 25, no. 4, pp. 63–73, 2011. View at Publisher · View at Google Scholar · View at Scopus - F. Qi, “A note on Schur-convexity of extended mean values,”
*Rocky Mountain Journal of Mathematics*, vol. 35, no. 5, pp. 1787–1793, 2005. View at Publisher · View at Google Scholar · View at Scopus - J.-L. Zhao, Q.-M. Luo, B.-N. Guo, and F. Qi, “Logarithmic convexity of Gini means,”
*Journal of Mathematical Inequalities*, vol. 6, no. 4, pp. 509–516, 2012. View at Publisher · View at Google Scholar - R. A. Satnoianu, “The proof of the conjectured inequality from the Mathematical Olympiad, Washington DC 2001,”
*Gazeta Matematica*, vol. 106, pp. 390–393, 2001. - W. Janous, “On a conjecture of Razvan Satnoianu,”
*Gazeta Matematica Seria A*, vol. 20, no. 99, pp. 97–103, 2002. - Y. Miao, S.-F. Xu, and Y.-X. Chen, “An answer to the conjecture of Satnoianu,”
*Applied Mathematics E-Notes*, vol. 9, pp. 262–265, 2009. - R. A. Satnoianu, “Improved GA-convexity inequalities,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 3, no. 5, article 82, 2002. - S.-H. Wu, “Note on a conjecture of R. A. Satnoianu,”
*Mathematical Inequalities and Applications*, vol. 12, no. 1, pp. 147–151, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. W. Marshall, I. Olkin, and B. C. Arnold,
*Inequalities: Theory of Majorization and Its Applications*, Springer, New York, NY, USA, 2nd edition, 2011. - Y.-M. Chu, X.-M. Zhang, and G.-D. Wang, “The schur geometrical convexity of the extended mean values,”
*Journal of Convex Analysis*, vol. 15, no. 4, pp. 707–718, 2008. View at Scopus