Cartwright's theorem

• Comment: Several claims here are incorrect, and the statement of the theorem is incoherent to the point of unreadability. When discussing this at WT:WPM, the draft author insisted "what think, sorry if I am wrong" without admitting the incorrectness of the claims. It cannot be accepted in its current form and needs significant cleanup (maybe a total rewrite) by someone who understands both the topic and English expository writing. —David Eppstein (talk) 19:12, 2 October 2022 (UTC)

• Comment: This draft currently says "for every integer $p\geq 1$, there exists a constant $C_{p}$ for any function $f(z)$" etc. The phrasing "there exists a constant $C_{p}$ for any function..." makes it sound as if the value of the constant depends on which function ƒ is. I suspect (but I'm not sure) that what was intended was "there exists a constant $C_{p}$ such that for every function $f(z)$". The notation "$|a_{i}|\max _{0\leq i\leq p}$" I am guessing was intended to be "$\max _{0\leq i\leq p}|a_{i}|$". Michael Hardy (talk) 03:55, 17 October 2022 (UTC)

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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 entire function $f(z)=\sum _{i=0}^{\infty }a_{n}z^{n}$ which is $p$-valent 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