On the sum of squared logarithms inequality and related inequalities
  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 $ 
Fozi M. Dannan and Patrizio Neff and Christian Thiel
math.CA