Real Analysis Exchange

Inequalities for Mean Values in Two Variables

Horst Alzer

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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.

Article information

Real Anal. Exchange, Volume 41, Number 1 (2016), 101-122.

First available in Project Euclid: 29 March 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26D07: Inequalities involving other types of functions
Secondary: 26A48: Monotonic functions, generalizations 26A51: Convexity, generalizations

Inequalities Mean values Completely monotonic


Alzer, Horst. Inequalities for Mean Values in Two Variables. Real Anal. Exchange 41 (2016), no. 1, 101--122.

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