Sharp bounds for the difference between the arithmetic and geometric means

We present sharp bounds for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sum_{i=1}^n \alpha_i x_i -\prod_{i=1}^n x_i^{\alpha_i} }$$\end{document} in terms of the variance of the vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(x_1^{1/2},\dots,x_n^{1/2})}$$\end{document}.