Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

Silvestru Sever Dragomir

doi:10.1556/314.2021.00011

Mathematica Pannonica

Volume/Issue:: Volume 27_NS1: Issue 2

Error Bounds Related to Midpoint and Trapezoid Rules for the Monotonic Integral Transform of Positive Operators in Hilbert Spaces

Author:

Silvestru Sever Dragomir

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:: 105–119

Online Publication Date:: 06 Oct 2021

Publication Date:: 18 Nov 2021

Article Category:: Research Article

DOI:: https://doi.org/10.1556/314.2021.00011

Keywords:: operator monotone functions; Operator convex functions; Operator inequalities; Midpoint inequality; Trapezoid inequality

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

M (w, μ) (T) := \int_{0}^{\infty} w (λ) T {(λ + T)}^{- 1} d μ (λ),

where the integral is assumed to exist forT a positive operator on a complex Hilbert spaceH. We show among others that, if β ≥ A, B ≥ α > 0, and 0 < δ ≤ (B − A)² ≤ Δ for some constants α, β, δ, Δ, then

0 \leq \frac{1}{24} δ M^{″} (w, μ) (β) \leq M (w, μ) (\frac{A + B}{2}) - \int_{0}^{1} M (w, μ) ((1 - t) A + t B) d t \leq - \frac{1}{24} Δ M^{″} (w, μ) (α)

and

0 \leq - \frac{1}{12} δ M^{″} (w, μ) (β) \leq \int_{0}^{1} M (w, μ) ((1 - t) A + t B) d t - \frac{M (w, μ) (A) + M (w, μ) (B)}{2} \leq \frac{1}{12} Δ M^{″} (w, μ) (α),

where $M^{″} (w, μ)$ is the second derivative of $M (w, μ)$ as a real function.

Applications for power function and logarithm are also provided.

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Mathematica Pannonica

Print ISSN:: 0865-2090

Online ISSN:: 2786-0752

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

Honorary Editors in Chief:

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

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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

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György GÁT, University of Debrecen, Debrecen, Hungary

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Mathematica Pannonica
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