SCHWEITZER 2015 Problem 9

For a function ${u}$ defined on ${G \subset \Bbb{C}}$ let us denote by ${Z(u)}$ the neignborhood of unit raduis of the set of roots of ${u}$ . Prove that for any compact set ${K \subset G}$ there exists a constant ${C}$ such that if ${u}$ is an arbitrary real harmonic function on ${G}$ which vanishes in a point of ${K}$ then: $$\sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.$$