Jensen's Inequality for Convex-Concave Antisymmetric Functions and Applications
© S. Hussain et al. 2008
Received: 21 February 2008
Accepted: 11 August 2008
Published: 9 September 2008
The weighted Jensen inequality for convex-concave antisymmetric functions is proved and some applications are given.
holds for every and every (see [1, Chapter 2]).
holds for every , where (see [1, page 7]).
Another way of weakening the assumptions for (1.1) is relaxing the assumption of positivity of weights . The most important result in this direction is the Jensen-Steffensen inequality (see, e.g., [1, page 57]) which states that (1.1) holds also if and , where .
The main purpose of this paper is to prove the weighted version of Theorem 1.1. For some related results, see [4, 5]. In Section 3, to illustrate the applicability of this result, we give a generalization of the famous Ky-Fan inequality.
2. Main Results
so the proof is complete.
Neither condition , nor condition , can be removed from the assumptions of Theorem 2.1. To see this, consider the function on . That the first condition cannot be removed can be seen by considering , and . That the second condition cannot be removed can be seen by considering , and . In both cases, (1.1) does not hold.
Recall that a function is 3-convex if for , and , where denotes third-order divided difference of . It is easy to prove, using properties of divided differences or using classical case of the Levinson inequality, that if is a 3-convex function, then the function is convex on (see [1, pages 71-72]).
where (see [6, page 295]).
In the following corollary, we give an improvement of the Ky-Fan inequality.
which by obvious rearrangement implies (3.6).
The research of J. Pečarić and I. Perić was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grants 117-1170889-0888 (J. Pečarić) and 058-1170889-1050 (I. Perić). S. Hussain and J. Pečarić also acknowledge with thanks the facilities provided to them by Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan. The authors also thank the careful referee for helpful suggestions which have improved the final version of this paper.
- Pečarić J, Proschan F, Tong YL: Convex Functions, Partial Orderings, and Statistical Applications, Mathematics in Science and Engineering. Volume 187. Academic Press, Boston, Mass, USA; 1992:xiv+467.MATHGoogle Scholar
- Hardy G, Littlewood JE, Polya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1967.MATHGoogle Scholar
- Mitrinović DS, Pečarić J: Generalizations of the Jensen inequality. Österreichische Akademie der Wissenschaften Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte 1987,196(1–3):21–26.MATHMathSciNetGoogle Scholar
- Czinder P: A weighted Hermite-Hadamard-type inequality for convex-concave symmetric functions. Publicationes Mathematicae Debrecen 2006,68(1–2):215–224.MATHMathSciNetGoogle Scholar
- Czinder P, Páles Z: An extension of the Hermite-Hadamard inequality and an application for Gini and Stolarsky means. Journal of Inequalities in Pure and Applied Mathematics 2004,5(2, article 42):-8.Google Scholar
- Bullen PS: Handbook of Means and Their Inequalities, Mathematics and Its Applications. Volume 560. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xxviii+537.View ArticleMATHGoogle Scholar
- Mercer AMcD: A variant of Jensen's inequality. Journal of Inequalities in Pure and Applied Mathematics 2003,4(4, article 73):-2.Google Scholar
- Pečarić J: On an inequality of N. Levinson. Publikacije Elektrotehničkog Fakulteta. Serija Matematika 1980, (678–715):71–74.MathSciNetMATHGoogle Scholar
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.