MathDB

a>0, f(a)<0 for f(\sqrt{xy})=\frac{f(x)+f(y)}{2}

Source: Mathcenter Contest / Oly - Thai Forum 2012 (R1) p5 sl-13 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 13, 2022

algebrainequalitiesfunctional

Problem Statement

Define

f : \mathbb{R}^+ \rightarrow \mathbb{R}

as the strictly increasing function such that

f(\sqrt{xy})=\frac{f(x)+f(y)}{2}

for all positive real numbers

x,y

. Prove that there are some positive real numbers

a

where f(a)<0.
(PP-nine)