Operator Convexity of an Integral Transform with Applications

Abstract

For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the following integral transformwhere the integral is assumed to exist for t > 0.We show among others that D(w, μ) is operator convex on (0, ∞). From this we derive that, if f : [0, ∞) → R is an operator monotone function on [0, ∞), then the function [f(0) -f(t)] t-1 is operator convex on (0, ∞). Also, if f : [0, ∞) → R is an operator convex function on [0, ∞), then the function is operator convex on (0, ∞). Some lower and upper bounds for the Jensen’s differenceunder some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.