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. ISSN 0862-9544 (Printed edition) Faculty of Mathematics, Physics and Informatics Comenius University 842 48 Bratislava, Slovak Republic Telephone: + 421-2-60295111 Fax: + 421-2-65425882 e-Mail: amuc@fmph.uniba.sk Internet: www.iam.fmph.uniba.sk/amuc © 2009, ACTA MATHEMATICA UNIVERSITATIS COMENIANAE |