INEQUALITIES FOR WEIGHTED ARITHMETIC AND GEOMETRIC MEANS

Abstract

Let

$${\mu }_n(\alpha ,\beta ;p,x)=\frac{\left(A_n\right(p,x){)}^{\alpha }}{\left(G_n\right(p,x){)}^{\beta }}(n\ge 2;\alpha ,\beta \in \mathrm{ℝ}),$$

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

(i) The inequality

$${\mu }_n(\alpha ,\beta ;p,x+y)\le {\mu }_n(\alpha ,\beta ;p,x)+{\mu }_n(\alpha ,\beta ;p,y)(*)$$

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

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

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