On log-sum inequalities

In information theory, the well-known log-sum inequality is a fundamental tool which indicates the non-negativity for the relative entropy. In this article, we establish a set of inequalities which are similar to the log-sum inequality involving two functions defined on scalars. The parametric extended log-sum inequalities are shown. We extend these inequalities for the commutative matrices. In addition, utilizing the L\"{o}wner partial order relation and the Hansen-Pedersen theory for non-commutative positive semi-definite matrices we demonstrate a number of matrix-inequalities analogous to the log-sum inequality.