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2021 INEQUALITIES FOR WEIGHTED ARITHMETIC AND GEOMETRIC MEANS
Horst Alzer
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Real Anal. Exchange 46(2): 359-366 (2021). DOI: 10.14321/realanalexch.46.2.0359

Abstract

Let

μn(α,β;p,x)=(An(p,x))α(Gn(p,x))β(n2;α,β),

where An(p,x) and Gn(p,x) denote the arithmetic and geometric means of x=(x1,,xn)+n with weights p=(p1,,pn)+n, p1++pn=1. We prove:

(i) The inequality

μn(α,β;p,x+y)μn(α,β;p,x)+μn(α,β;p,y)(*)

is valid for all x,y+n and p+n with p1++pn=1 if and only if βmax(α1,0).

(ii) Inequality (*) with “” instead of “” holds for all x,y+n and p+n with p1++pn=1 if and only if α0 and βmax(α1,0).

This extends a result of Dragomir, Comănescu and Pearce, who proved (*) for the special case α=2, β=1.

Citation

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Horst Alzer. "INEQUALITIES FOR WEIGHTED ARITHMETIC AND GEOMETRIC MEANS." Real Anal. Exchange 46 (2) 359 - 366, 2021. https://doi.org/10.14321/realanalexch.46.2.0359

Information

Published: 2021
First available in Project Euclid: 8 November 2021

Digital Object Identifier: 10.14321/realanalexch.46.2.0359

Subjects:
Primary: 26D07, ‎39B62

Rights: Copyright © 2021 Michigan State University Press

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Vol.46 • No. 2 • 2021
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