Cartwright's theorem

In mathematics, specially in complex analysis, Cartwright's theorem, first discovered by British mathematician Mary Cartwright, gives an estimate of an analytical function's maximum modulus when the unit disc takes the same value no more than p times..

Statement

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

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

in an absolute value for all $z$ in the disc $|z|\leq r$ and $r\leq 1$.

Further reading

• https://www.theoremoftheday.org/Analysis/Cartwright/TotDCartwright.pdf