MathDB

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:

\displaystyle \sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.

Problem Statement