Cartwright's theorem

In mathematics, Cartwright's Theorem belongs to Graph Theory and was first discovered by British Mathematician Mary Cartwright. This theorem gives an estimate of an analytical function's maximum modulus when the unit disc takes the same value no more than p times. This theorem has applications of other mathematical concepts such as Set theory.

Statement

Cartwright's Theorem says that, for every integer $p$ such that $p\geq 1$, there exists a constant $C_{p}$ for any function $f(z)$ which is $p$ - valent in disc $|z|<1$, analytic, and expressible in series form of $f(z)=\Sigma _{i=0}^{\infty }a_{n}z^{n}$, which is bounded as $|f(z)|\leq {\frac {|a_{i}|max_{0\leq i\leq p}}{(1-r)^{2p}}}C_{p}$ in an absolute value for all $z$ in the disc $|z|\leq r$ and $r\leq 1$.