Mean Value Inequalities

descriptionPublicationkeyboard_double_arrow_right Article 01 Oct 1977 English Publisher:Society for Industrial & Applied Mathematics (SIAM)Journal:SIAM Journal on Mathematical Analysis, volume 8, pages 785-791 (issn: 0036-1410, eissn: 1095-7154,

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Authors: Fink, A. M.;

doi: 10.1137/0508061

Mean Value Inequalities

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Abstract

We find the best possible constants $K_i = K_i (\varphi )$, $i = 1,2$, for inequalities of the kind \[f(t)\int_0^t {\varphi (f(s))ds \leqq K_i \varphi (f(t))} \int_0^t {f(s)ds} \] when $\varphi $ is a given positive function, valid for all functions f such that $f(0) = 0$ and either $(i = 2)f$ is increasing and convex, or $(i = 2)f$ is increasing.

Keywords

Inequalities for sums, series and integrals, Convexity of real functions in one variable, generalizations

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