J. oflnequal. & Appi., 2001, Vol. 6, pp. 451-462 Reprints available directly from the publisher Photocopying permitted by license only (C) 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint, a member of the Taylor & Francis Group. An Extension of Chebyshev’s Inequality and its Connection with Jensen’s Inequality* CONSTANTIN P. NICULESCU Department of Mathematics, University of Craiova, Craiova 1100, Romania (Received 24 November 1999; In final form 17 February 2000) The aim of this paper is to show that Jensen’s Inequality and an extension of Chebyshev’s Inequality complement one another, so that they both can be formulated in a pairing form, including a second inequality, that provides an estimate for the classical one. Keywords: Convex functions; Subdifferential; Mean value Mathematics Subject Classifications 2000: Primary: 26A51, 26D15 1. INTRODUCTION The well known fact that the derivative and the integral are inverse each other has a lot of interesting consequences, one of them being the duality between convexity and monotonicity. The purpose of the present paper is to relate on this basis two basic inequalities in Classical Analysis, precisely those due to Jensen and Chebyshev. Both refer to mean values of integrable functions. Restricting ourselves to the case of finite measure spaces (X, , #), let us recall * Partially supported by MEN Grant 39683/1998. e-mail: tempus@oltenia.ro 451