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.
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