MathDB

Let

\mathbb{R}_{>0}

be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions

f:\mathbb{R}_{>0} \to \mathbb{R}

such that there exists a two-variable polynomial

P(x, y)

with real coefficients satisfying

f(xy)=P(f(x), f(y))

for all

x, y\in\mathbb{R}_{>0}

.\\
Proposed by Navid Safaei, Iran

Problem Statement