Generalization and reverses of the left Fejer inequality for convex functions

In this paper we establish a generalization of the left Fejer inequality for general Lebesgue integral on measurable spaces as well as various upper bounds for the difference 1 / ʃ b a g (x) dx ʃ b a h (x) g (x) dx − h ( a + b / 2 ) , where h : [a, b] → R is a convex function and g : [a, b] → [0, ∞) is an integrable weight. Applications for discrete means and Hermite-Hadamard type inequalities are also provided.