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Autumn 2018 Estimations of the Lehmer mean by the Heron mean and their generalizations involving refined Heinz operator inequalities
Masatoshi Ito
Adv. Oper. Theory 3(4): 763-780 (Autumn 2018). DOI: 10.15352/aot.1801-1303

Abstract

As generalizations of the arithmetic and the geometric means, for positive real numbers $a$ and $b$, the power difference mean $J_{q}(a,b)=\frac{q}{q+1}\frac{a^{q+1}-b^{q+1}}{a^{q}-b^{q}}$, the Lehmer mean $L_{q}(a,b)=\frac{a^{q+1}+b^{q+1}}{a^{q}+b^{q}}$ and the Heron mean $K_{q}(a,b)=(1-q)\sqrt{ab}+q\frac{a+b}{2}$ are well known.

In this paper, concerning our recent results on estimations of the power difference mean, we obtain the greatest value $\alpha=\alpha(q)$ and the least value $\beta=\beta(q)$ such that the double inequality for the Lehmer mean $$K_{\alpha}(a,b)< L_{q}(a,b)< K_{\beta}(a,b)$$ holds for any $q \in \mathbb{R}$. We also obtain an operator version of this estimation. Moreover, we discuss generalizations of the results on estimations of the power difference and the Lehmer means.This argument involves refined Heinz operator inequalities by Liang and Shi.

Citation

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Masatoshi Ito. "Estimations of the Lehmer mean by the Heron mean and their generalizations involving refined Heinz operator inequalities." Adv. Oper. Theory 3 (4) 763 - 780, Autumn 2018. https://doi.org/10.15352/aot.1801-1303

Information

Received: 25 January 2018; Accepted: 24 April 2018; Published: Autumn 2018
First available in Project Euclid: 10 May 2018

zbMATH: 06946376
MathSciNet: MR3856171
Digital Object Identifier: 10.15352/aot.1801-1303

Subjects:
Primary: 47A63
Secondary: 26E60, 47A64

Rights: Copyright © 2018 Tusi Mathematical Research Group

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Vol.3 • No. 4 • Autumn 2018
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