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

Abstract

Let

$\mu_n(\alpha,\beta;p,x)=\frac{(A_n(p,x))^\alpha}{(G_n(p,x))^\beta}(n\geq2;\alpha,\beta\in\mathbb{R}),$

where $A_n(p,x)$ and $G_n(p,x)$ denote the arithmetic and geometric means of $x=(x_1,\dots,x_n)\in\mathbb{R}_+^n$ with weights $p=(p_1,\dots,p_n)\in\mathbb{R}_+^n$, $p_1+\cdots+p_n=1$. We prove:

(i) The inequality

$\mu_n(\alpha,\beta;p,x+y)\leq\mu_n(\alpha,\beta;p,x)+\mu_n(\alpha,\beta;p,y)(\ast)$

is valid for all $x,y\in\mathbb{R}_+^n$ and $p\in\mathbb{R}_+^n$ with $p_1+\cdots+p_n=1$ if and only if $\beta\geq\max(\alpha-1,0)$.

(ii) Inequality  with “$\geq$” instead of “$\leq$” holds for all $x,y\in\mathbb{R}_+^n$ and $p\in\mathbb{R}_+^n$ with $p_1+\cdots+p_n=1$ if and only if  and $\beta\leq\max(\alpha-1,0)$.

This extends a result of Dragomir, Comănescu and Pearce, who proved  for the special case $\alpha=2$, $\beta=1$.

Citation

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  