Cartwright's theorem

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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 to other mathematical concepts such as set theory.

Statement

Cartwright's theorem states that, for every integer $p\ge 1$, there exists a constant $C_p$ such that for any function $f(z)$ which is $p$-valent in disc $|z|<1$, analytic, and expressible in series form as $f(z)=\sum _{i=0}^{\mathrm{\infty }}{a}_n{z}^n$, it is bounded as $|f(z)|\le \frac{\underset{0\le i\le p}{\max }|a_i|}{(1-r{)}^{2p}}{C}_p$ in absolute value for all $z$ in the disc $|z|\le r$ and $r\le 1$.

References

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