Communications in Mathematical Analysis

On Discrete Favard's and Berwald's Inequalities

N. Latif, I. Perić, and J. Pečarić

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Abstract

In this paper, we obtain an extensions of majorization type results and extensions of weighted Favard's and Berwald's inequality. We prove positive semi-definiteness of matrices generated by differences deduced from majorization type results and differences deduced from weighted Favard's and Berwald's inequality. This implies a surprising property of exponentially convexity and $\log$-convexity of this differences which allows us to deduce Lyapunov's inequalities for the differences, which are improvements of majorization type results and weighted Favard's and Berwald's inequalities. Analogous Cauchy's type means, as equivalent forms of exponentially convexity and log-convexity, are also studied and the monotonicity properties are proved.

Article information

Source
Commun. Math. Anal., Volume 12, Number 2 (2012), 34-57.

Dates
First available in Project Euclid: 16 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.cma/1331929870

Mathematical Reviews number (MathSciNet)
MR2905130

Zentralblatt MATH identifier
1263.26008

Subjects
Primary: 26A24
Secondary: 26A51 26D15: Inequalities for sums, series and integrals

Keywords
Convex function majorization generalized Favard's inequality positive semi-definite matrix exponential convexity log-convexity Lypunov's inequality Dresher's type of means Cauchy means

Citation

Latif, N.; Pečarić, J.; Perić, I. On Discrete Favard's and Berwald's Inequalities. Commun. Math. Anal. 12 (2012), no. 2, 34--57. https://projecteuclid.org/euclid.cma/1331929870


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