Author:
Silvestru Sever Dragomir Mathematics, College of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia
DST-NRF Centre of Excellence in the Mathematical, and Statistical Sciences, School of Computer Science & Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa

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For a continuous and positive function w(λ), λ > 0 and μ a positive measure on (0, ∞) we consider the following integral transform

D w , μ t : = 0 w λ λ + t 1 d μ λ ,

where 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 f 0 + f + 0 t f t t 2 is operator convex on (0, ∞). Some lower and upper bounds for the Jensen’s difference

D w , μ A + D w , μ B 2 D w , μ A + B 2

under some natural assumptions for the positive operators A and B are given. Examples for power, exponential and logarithmic functions are also provided.

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Mathematica Pannonica
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