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.