Problem

Source: Iranian third round 2019 Finals algebra exam problem 3

Tags: function, functional equation, algebra



Let $a,b,c$ be non-zero distinct real numbers so that there exist functions $f,g:\mathbb{R}^{+} \to \mathbb{R}$ so that: $af(xy)+bf(\frac{x}{y})=cf(x)+g(y)$ For all positive real $x$ and large enough $y$. Prove that there exists a function $h:\mathbb{R}^{+} \to \mathbb{R}$ so that: $f(xy)+f(\frac{x}{y})=2f(x)+h(y)$ For all positive real $x$ and large enough $y$.