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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Some improvements of the young mean inequality and its reverse

Maryam Khosravi
• Department of Pure Mathematics Faculty of Mathematics and Computer Shahid Bahonar University of Kerman, Kerman, Iran
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Published Online: 2018-08-06 | DOI: https://doi.org/10.1515/ms-2017-0146

Abstract

The main objective of the present paper, is to obtain some new versions of Young-type inequalities with respect to two weighted arithmetic and geometric means and their reverses, using two inequalities

$K(ba,2)r≤a∇νba♯νb≤K(ba,2)R,$

where r = min{ν, 1 – ν}, R = max{ν,1 – ν} and K(t,2) = $\begin{array}{}\frac{\left(t+1{\right)}^{2}}{4t}\end{array}$ is the Kantorovich constant, and

$e(h−1,ν)≤a∇νba♯νb≤e(h,ν),$

where h = max $\begin{array}{}\left\{\frac{a}{b},\frac{b}{a}\right\}\end{array}$ and e(t,ν) = exp (4ν(1 – ν)(K(t,2)–1) $\begin{array}{}\left(1-\frac{1}{2t}\right)\right).\end{array}$ Also some operator versions of these inequalities and some inequalities related to Heinz mean are proved.

MSC 2010: Primary 47A63; Secondary 47A64; 47B65

References

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Accepted: 2017-01-14

Published Online: 2018-08-06

Published in Print: 2018-08-28

Communicated by Werner Timmermann

Citation Information: Mathematica Slovaca, Volume 68, Issue 4, Pages 803–810, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918,

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