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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Furuta, T. Precise lower bound of f(A) - f(B) for A > B > 0 and non-constant operator monotone function f on (0, ∞). J. Math. Inequal., 9, 1 (2015), 47-52.

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Furuta, T. , Hot, T. M. , Pecarić J. and Seo, Y . Mond-Pecarić Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space. Element, Zagreb, 2005.

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Moslehian, M. S. and Najafi, H. An extension of the Löwner-Heinz inequality. Linear Algebra Appl, 437 (2012), 2359-2365.

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Generalized Exponential Integral. Digital Library of Mathematical Functions. NIST. https://dlmf.nist.gov/8.19#E1

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Editor in Chief: László TÓTH (University of Pécs)

Honorary Editors in Chief:

• † István GYŐRI (University of Pannonia, Veszprém)
• János PINTZ (Rényi Institute of Mathematics)
• Ferenc SCHIPP (Eötvös University Budapest and University of Pécs)
• Sándor SZABÓ (University of Pécs)

Deputy Editors in Chief:

• Erhard AICHINGER (JKU Linz)
• Ferenc HARTUNG (University of Pannonia, Veszprém)
• Ferenc WEISZ (Eötvös University, Budapest)

Editorial Board

• György DÓSA (University of Pannonia, Veszprém)
• István BERKES (Rényi Institute of Mathematics)
• Károly BEZDEK (University of Calgary)
• Balázs KIRÁLY – Managing Editor (University of Pécs)
• Vedran KRCADINAC (University of Zagreb)
• Željka MILIN ŠIPUŠ (University of Zagreb)
• Margit PAP (University of Pécs)
• Mihály PITUK (University of Pannonia, Veszprém)
• Jörg THUSWALDNER (Montanuniversität Leoben)
• Zsolt TUZA (University of Pannonia, Veszprém)

• Szilárd RÉVÉSZ (Rényi Institute of Mathematics)  - Chair
• György GÁT (University of Debrecen)

University of Pécs,
Faculty of Sciences,
Institute of Mathematics and Informatics
Department of Mathematics
7624 Pécs, Ifjúság útja 6., HUNGARY
(36) 72-503-600 / 4179
ltoth@gamma.ttk.pte.hu

• Mathematical Reviews
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Mathematica Pannonica
Language English
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