Gauss’ and Related Inequalities

Abstract

Let $g : [a,b] \to \mathbb R$ be a non-negative increasing differentiable function and $f : [a,b] \to \mathbb R$ a non-negative function such that the quotient $f/g’$ is non-decreasing. Then the function

is log-concave. If $g(a) = 0, b \in (a, \infty]$ and the quotient $f/g’$ is non-increasing, then the function $Q$ is log-convex.