paper

On the sum of squared logarithms inequality and related inequalities

arXiv:1411.1290

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 $a, b \in R_{+}^{n}$ so that $\sum_{i = 1}^{n} (lo g a_{i})^{2} \leq \sum_{i = 1}^{n} (lo g b_{i})^{2} .$ Generalizations of some inequalities from information theory are obtained, including a generalized information inequality and a generalized log sum inequality, which states for $a, b \in R_{+}^{n}$ and $k_{1}, ..., k_{n} \in [0, \infty)$ : $\sum_{i = 1}^{n} a_{i} lo g \prod_{s = 1}^{m} (\frac{a _{i}}{b _{i}} + k_{s}) \geq lo g \prod_{s = 1}^{m} (1 + k_{s}) .$