Inequalities for Mean Values in Two Variables

Abstract

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.