Comparing two integral means for absolutely continuous functions whose absolute value of the derivative are convex and applications

Some new estimates for the di¤erence between the integral mean of a function and its mean over a subinterval are established and new applications for special means and probability density functions are also given. 1. Introduction The classical Ostrowski type integral inequality [1] stipulates a bound for the di¤erence between a function evaluated at an interior point and the average of the function over an interval. That is, (1.1) 1 b a Z b a f(x)dx f(x) 1 4 + (x a+b 2 ) (b a)2 (b a)kf k1 for all x 2 [a; b]; where f 0 2 L1(a; b), that is, kf k1 = ess sup t2[a;b] jf 0(t)j <1; and f : [a; b] ! R is a di¤erentiable function on (a; b). Here, the constant 14 is sharp in the sense that it cannot be replaced by a smaller constant. For various results and generalizations concerning Ostrowski’s inequality, see [2-13] and the references therein. In [14], Barnett et al. compared the di¤erence of two integral means as in the following Theorem 1 in which the function has the …rst derivative bounded where is de…ned. The obtained results are also generalizations of (1.1) and have been applied to probability density functions, special means, Je¤reys divergence in Information Theory and the sampling of continuous streams in Statistics. Theorem 1. Let f : [a; b] ! R be an absolutely continuous function with the property that f 0 2 L1[a; b]. Then, for a x < y b, we have the inequality 1 b a Z b