Open Access
2016 Inequalities for Mean Values in Two Variables
Horst Alzer
Real Anal. Exchange 41(1): 101-122 (2016).


We present various inequalities for means in two variables. One of our results states that the inequalities $$ 0\leq \frac{1}{M_r} -\frac{1}{M_s} \leq \frac{1}{G}-\frac {1}{A } \quad{(r,s\geq 0)} $$ hold for all $x,y>0$ if and only if $0\leq s-r\leq 1$. Here, $A=A(x,y)=(x+y)/2$, $G=G(x,y)=\sqrt{xy}$ and $M_t=M_t(x,y)=[(x^t+y^t)/2]^{1/t}$ denote the arithmetic, geometric and power mean of $x$ and $y$, respectively.


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Horst Alzer. "Inequalities for Mean Values in Two Variables." Real Anal. Exchange 41 (1) 101 - 122, 2016.


Published: 2016
First available in Project Euclid: 29 March 2017

zbMATH: 06848918
MathSciNet: MR3511938

Primary: 26D07
Secondary: 26A48 , 26A51‎

Keywords: completely monotonic , Inequalities‎ , mean values

Rights: Copyright © 2016 Michigan State University Press

Vol.41 • No. 1 • 2016
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