Log-concavity of Hölder means and an application to geometric inequalities

Abstract

The log-concavity of the Hölder mean of two numbers, as a function of its index, is presented first. The notion of $\alpha$-cevian of a triangle is introduced next, for any real number $\alpha$. We use this property of the Hölder mean to find the smallest index $p\left(\alpha \right)$ such that the length of an $\alpha$-cevian of a triangle is less than or equal to the $p\left(\alpha \right)$-Hölder mean of the lengths of the two sides of the triangle that are adjacent to that cevian.