Problem

Source: Iranian National Olympiad (3rd Round) 2008

Tags: function, complex analysis, complex analysis unsolved



Let $ g,f: \mathbb C\longrightarrow\mathbb C$ be two continuous functions such that for each $ z\neq 0$, $ g(z)=f(\frac1z)$. Prove that there is a $ z\in\mathbb C$ such that $ f(\frac1z)=f(-\bar z)$