Mathematica Pannonica
Volume/Issue:
Accepted Manuscript / Online First
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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Online Publication Date:
17 Oct 2022
Publication Date:
17 Oct 2022
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

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.

  • [1]

    Bhatia, R . Matrix Analysis. Graduate Texts in Mathematics, 169. Springer-Verlag, New York, 1997. xii+347 pp. ISBN: 0-387-94846-5.

  • [2]

    Choi, M. D. Positive linear maps on C*-algebras. Canad. J. Math., 24 (1972), 520-529.

  • [3]

    Fujii, J. I. and Seo, Y. On parametrized operator means dominated by power ones. Sci. Math., 1 (1998) 301-306.

  • [4]

    Furuta, T . Concrete examples of operator monotone functions obtained by an elementary method without appealing to Löwner integral representation. Linear Algebra Appl., 429 (2008) 972-980.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [5]

    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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [6]

    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.

    • Search Google Scholar
    • Export Citation
  • [7]

    Heinz, E. Beitrage zur Störungsteorie der Spektralzerlegung. Math. Ann., 123 (1951) 415-438.

  • [8]

    Löwner K . Über monotone MatrixFunktionen. Math. Z, 38 (1934) 177-216.

  • [9]

    Moslehian, M. S. and Najafi, H. Around operator monotone functions. Integr. Equ. Oper. Theory, 71 (2011), 575-582.

  • [10]

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

  • [11]

    Zuo, H ., Duan, G. Some inequalities of operator monotone functions. J. Math. Inequal., 8, 4 (2014), 777-781.

  • [12]

    Incomplete Gamma and Related Functions, Definitions. Digital Library of Mathematical Functions. NIST. https://dlmf.nist.gov/8.2

  • [13]

    Incomplete Gamma and Related Functions, Integral Representations. Digital Library of Mathematical Functions. NIST. https://dlmf.nist.gov/8.6

    • Search Google Scholar
    • Export Citation
  • [14]

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

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

Advisory Board

  • Szilárd RÉVÉSZ (Rényi Institute of Mathematics)  - Chair
  • Gabriella BÖHM (Akadémiai Kiadó, Budapest)
  • György GÁT (University of Debrecen)

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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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ISSN 2786-0752 (Online)
ISSN 0865-2090 (Print)

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