Elementary inequalities that involve two nonnegative vectors or functions

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National Academy of Sciences

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We report 96 inequalities with common structure, all elementary to state but many not elementary to prove. If n is a positive integer, a = (a1,..., an) and b = (b1,..., bn) are arbitrary vectors in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*}R_{+}^{n}=[0,{\infty})^{n}\end{equation*}\end{document}, and ρ(mij) is the spectral radius of an n × n matrix with elements mij, then, for example: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} \begin{equation*} \begin{matrix}\\ {{\sum_{i,j}}}{\mathrm{min}}((a_{i}a_{j}),(b_{i}b_{j})){\leq}{{\sum_{i,j}}}{\mathrm{min}}((a_{i}b_{j}),(b_{i}a_{j})),\\ {{\sum_{i,j}}}{\mathrm{max}}((a_{i}+a_{j}),(b_{i}+b_{j})){\geq}{{\sum_{i,j}}}{\mathrm{max}}((a_{i}+b_{j}),(b_{i}+a_{j})),\\ {\rho}({\mathrm{min}}((a_{i}a_{j}),(b_{i}b_{j}))){\leq}{\rho}({\mathrm{min}}((a_{i}b_{j}),(b_{i}a_{j}))),\\ {{\sum_{i,j}}}{\mathrm{min}}((a_{i}a_{j}),(b_{i}b_{j}))x_{i}x_{j}{\leq}{{\sum_{i,j}}}{\mathrm{min}}((a_{i}b_{j}),(b_{i}a_{j}))x_{i}x_{j},{\mathrm{for\;all\;real}}~x_{i},i=1,{\ldots},n,\\ {\int \int }{\mathrm{log}}[(f(x)+f(y))(g(x)+g(y))]d{\mu}(x)d{\mu}(y){\leq}{\int \int }{\mathrm{log}}[(f(x)+g(y))(g(x)+f(y))]d{\mu}(x)d{\mu}(y).\end{matrix} \end{equation*}\end{document} The second inequality is obtained from the first inequality by replacing min with max and × with + and by reversing the direction of the inequality. The third inequality is obtained from the first by replacing the summation by the spectral radius. The fourth inequality is obtained from the first by taking each summand as a coefficient in a quadratic form. The fifth inequality is obtained from the first by replacing both outer summations by products, min by ×, × by +, and the nonnegative vectors a and b by nonnegative measurable functions f and g. The proofs of these inequalities are mysteriously diverse.

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