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arXiv · 1411.1290

On the sum of squared logarithms inequality and related inequalities

Fozi M. Dannan·Patrizio Neff·Christian Thiel
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Abstract

We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors $a,b\in\mathbb{R}_+^n$ so that $\sum_{i=1}^n(\log a_i)^2\ \leq\ \sum_{i=1}^n(\log b_i)^2\,.\notag $ Generalizations of some inequalities from information theory are obtained, including a generalized information inequality and a generalized log sum inequality, which states for $a,b\in\mathbb{R}_+^n$ and $k_1,...,k_n\in [0,\infty)$: $ \sum_{i=1}^na_i\,\log\prod_{s=1}^m(\frac{a_i}{b_i} + k_s)\ \geq\ \log\prod_{s=1}^m(1+k_s)\,.\notag $

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Fozi M. Dannan, Patrizio Neff, Christian Thiel. 2014-11-05. On the sum of squared logarithms inequality and related inequalities. https://arxiv.org/abs/1411.1290

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