Mathematica Pannonica
Volume/Issue:
Volume 29_NS3: Issue 1
Operator Convexity of an Integral Transform with Applications
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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Pages:
17–28
Online Publication Date:
17 Oct 2022
Publication Date:
19 May 2023
Article Category:
Research Article
DOI:
https://doi.org/10.1556/314.2022.00024
Keywords:
Operator monotone functions; operator convex functions; operator inequalities; Löwner-Heinz inequality; logarithmic operator inequalities
Open access

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

Dw,μt:=∫0∞wλλ+t−1dμλ,

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 f0+f+′0t−ftt−2 is operator convex on (0, ∞). Some lower and upper bounds for the Jensen’s difference

Dw,μA+Dw,μB2−Dw,μA+B2

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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Cover Mathematica Pannonica
Mathematica Pannonica
Print ISSN:
0865-2090
Online ISSN:
2786-0752

Editor in Chief: László TÓTH, University of Pécs, Pécs, Hungary

Honorary Editors in Chief:

  • † István GYŐRI, University of Pannonia, Veszprém, Hungary
  • János PINTZ, HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Ferenc SCHIPP, Eötvös Loránd University, Budapest, Hungary and University of Pécs, Pécs, Hungary
  • Sándor SZABÓ, University of Pécs, Pécs, Hungary
     

Deputy Editors in Chief:

  • Erhard AICHINGER, JKU Linz, Linz, Austria
  • Ferenc HARTUNG, University of Pannonia, Veszprém, Hungary
  • Ferenc WEISZ, Eötvös Loránd University, Budapest, Hungary

Editorial Board

  • Attila BÉRCZES, University of Debrecen, Debrecen, Hungary
  • István BERKES, Rényi Institute of Mathematics, Budapest, Hungary
  • Károly BEZDEK, University of Calgary, Calgary, Canada
  • György DÓSA, University of Pannonia, Veszprém, Hungary
  • Balázs KIRÁLY – Managing Editor, University of Pécs, Pécs, Hungary
  • Vedran KRCADINAC, University of Zagreb, Zagreb, Croatia 
  • Željka MILIN ŠIPUŠ, University of Zagreb, Zagreb, Croatia
  • Gábor NYUL, University of Debrecen, Debrecen, Hungary
  • Margit PAP, University of Pécs, Pécs, Hungary
  • István PINK, University of Debrecen, Debrecen, Hungary
  • Mihály PITUK, University of Pannonia, Veszprém, Hungary
  • Lukas SPIEGELHOFER, Montanuniversität Leoben, Leoben, Austria
  • Andrea ŠVOB, University of Rijeka, Rijeka, Croatia
  • Csaba SZÁNTÓ, Babeş-Bolyai University, Cluj-Napoca, Romania
  • Jörg THUSWALDNER, Montanuniversität Leoben, Leoben, Austria
  • Zsolt TUZA, University of Pannonia, Veszprém, Hungary

Advisory Board

  • Szilárd RÉVÉSZ – Chair, HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • Gabriella BÖHM
  • György GÁT, University of Debrecen, Debrecen, Hungary

University of Pécs,
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Department of Mathematics
7624 Pécs, Ifjúság útja 6., HUNGARY
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ltoth@gamma.ttk.pte.hu

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Mathematica Pannonica
Language English
Size A4
Year of
Foundation		 1990
Volumes
per Year		 1
Issues
per Year		 2
Founder Akadémiai Kiadó
Founder's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Publisher Akadémiai Kiadó
Publisher's
Address		 H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
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Publisher		 Chief Executive Officer, Akadémiai Kiadó
ISSN 2786-0752 (Online)
ISSN 0865-2090 (Print)

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