Cartwright's theorem

Cartwright's theorem is a mathematical theorem in complex analysis, discovered by the British mathematician Mary Cartwright. It gives an estimate of the maximum modulus of an analytical function when the unit disc takes the same value no more than p times.^[1]

Statement

Cartwright's theorem says that, for every integer ${\displaystyle p\geq 1}$, there exists a constant ${\displaystyle C_{p}}$ such that for every ${\displaystyle p}$-valent holomorphic function ${\displaystyle f(z)=\sum _{i=0}^{\infty }a_{n}z^{n}}$ in disc ${\displaystyle |z|<1}$, we have the bound

${\displaystyle |f(z)|\leq {\frac {\max _{0\leq i\leq p}|a_{i}|}{(1-r)^{2p}}}C_{p}}$

in an absolute value for all ${\displaystyle z}$ in the disc ${\displaystyle |z|\leq r}$ and ${\displaystyle r\leq 1}$.^[2][3]