Erdős-Mordell-Type Inequalities in a Triangle

with equality if and only if the triangle is equilateral and P is its center. This inequality was conjectured by Erdős [1] and proved by Mordell and Barrow [2]. Oppenheim [3] established a number of additional inequalities relating the six distances p, q, r , x , y, and z. Such an inequality will be referred to as an Erdős-Mordell-type inequality. A survey of some of these inequalities can be found in [4]. The aim of this note is to obtain a large family of Erdős-Mordell-type inequalities. To do so we first note that inequality (1) compares the elements f (p), f (q), f (r) with f (x), f (y), f (z) for the function f (t) = t . By applying one of the standard proofs for (1) (for example, as in [4]) and replacing f with a monotone multiplicatively convex function, a large class of Erdős-Mordell-type inequalities is obtained. Montel was among the first to consider multiplicative convexity [5]. A modern presentation is given by Niculescu [6]. We begin by recalling the definition of this concept.