An Extension of Chebyshevs Inequality and Its Connection with Jensens Inequality
نویسنده
چکیده
The aim of this paper is to show that Jensens Inequality and an extension of Chebyshevs 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. 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 nite measure spaces (X; ; ), let us recall that the mean value of any integrable function f : X ! R can be de ned as
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