ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

A Functional Generalization of Ostrowski inequality
via Montgomery identity

S. S. Dragomir

Received: March 17, 2014;   Accepted: July 2, 2014

Abstract.   We show in this paper amongst other that, if f: [a, b]\to R is absolutely continuous on [a, b] and \Phi: R\to R is convex (concave) on R then
$$\Phi(f(x)-\frac{1}{b-a}\int_{a}^{b}f(t)dt) \leq (\geq) \frac{1}{b-a}\left[\int_{a}^{x}\Phi [(t-a)f'(t)] dt +\int_{x}^{b}\Phi[( t-b) f'(t)] dt]$$
for any x\in [a, b]. Natural applications for power and exponential functions are provided as well. Bounds for the Lebesgue p-norms of the deviation of a function from its integral mean are also given.

Keywords:  Absolutely continuous functions, Convex functions, Integral inequalities, Ostrowski inequality, Jensen's inequality, Lebesgue norms, Special means.

AMS Subject classification: Primary:  26D15, 25D10.

Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic

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