View source for Draft:Cartwright's theorem

{{AFC submission|d|reason|Echoing the comment by David Eppstein below, the draft as it correctly stands is unintelligible. |u=Aitzaz Imtiaz|ns=118|decliner=Felix QW|declinets=20221014081546|ts=20221001231749}} <!-- Do not remove this line! --> 

{{AFC comment|1=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. —[[User:David Eppstein|David Eppstein]] ([[User talk:David Eppstein|talk]]) 19:12, 2 October 2022 (UTC)}}

{{AFC comment|1=This draft currently says "for every integer <math>p \ge 1</math>, there exists a constant <math>C_p</math> for any function <math>f(z)</math>" etc. The phrasing "there exists a constant <math>C_p</math> for any function..." makes it sound as if the value of the constant depends on which function ''&fnof;'' is. I suspect (but I'm not sure) that what was intended was "there exists a constant <math>C_p</math> such that for every function <math>f(z)</math>". The notation "<math>|a_i|\max_{0 \leq i \leq p}</math>" I am guessing was intended to be "<math>\max_{0 \leq i \leq p} |a_i|</math>". [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 03:55, 17 October 2022 (UTC)}}

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{{Short description|Cartwright's theorem}}
{{Draft topics|computing}}
{{AfC topic|stem}}

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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 of other mathematical concepts such as [[set theory]]<ref>{{Cite journal |last1=Liu |first1=H. C. |last2=Macintyre |first2=A. J. |date= |title=CARTWRIGHT'S THEOREM ON FUNCTIONS BOUNDED AT THE INTEGERS |url=https://www.ams.org/journals/proc/1961-012-03/S0002-9939-1961-0125222-4/S0002-9939-1961-0125222-4.pdf |journal=American Mathematical Society |doi=}}</ref>. 

== Statement ==
Cartwright's theorem says that, for every integer <math>p \ge 1</math>, there exists a constant <math>C_p</math> for any function <math>f(z)</math> which is <math>p</math>-valent in disc <math>|z| < 1</math>, analytic, and expressible in series form of <math>f(z) = \sum_{i=0}^\infty a_n z^n</math>, which is bounded as <math>|f(z)| \leq \frac{\max_{0 \leq i \leq p}|a_i|}{(1-r)^{2p}} C_p</math> in an absolute value for all <math>z</math> in the disc <math>|z| \leq r</math> and <math>r \leq 1</math>.<ref>{{Cite journal |last1=Blank |first1=Natalia |last2=Ulanovskii |first2=Alexander |title=On Cartwright's theorem |date=October 2016 |url=https://www.ams.org/journals/proc/2016-144-10/S0002-9939-2016-13200-0/S0002-9939-2016-13200-0.pdf |journal=Proceedings of the American Mathematical Society |volume=144 |issue=10 |pages=4221–4230|doi=10.1090/proc/13200 |s2cid=119148466 }}</ref><ref>{{Cite web |last1=McMurran |first1=Shawnee |last2=Tattersall |first2=James |title=Mary Cartwright |url=https://www.ams.org/notices/199902/mem-cartwright.pdf |website=American Mathematical Society}}</ref>

== References ==
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