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]).
Let be Wright-convex on and . If and for , then (1.4) is valid.
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
Let be a convex function on and for every . If , and for , then (1.1) holds.
Since the function is convex on and , by Galvani's theorem it follows that the function is increasing on . Therefore, from and we have ; so (2.4) holds.
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.
where is a convex function on , and for . If is concave, the reverse inequalities hold in (2.7).
which is the reverse of the right-hand side inequality of (2.7); so the concavity properties of the function are prevailing in this case.
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]).
It is a simple consequence of Theorem 2.1 and the above-mentioned fact that is convex on .
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.
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