Goodman's conjecture

Let $f(z)=\; \backslash sum\_\{n=1\}^\{\backslash infty\}\{b\_n\; z^n\}$ be a $p$-valent function. The conjecture claims the following coefficients hold:

$$|b\_n|\; \backslash le\; \backslash sum\_\{k=1\}^\{p\}\; \backslash frac\{2k(n+p)!\}\{(p-k)!(p+k)!(n-p-1)!(n^2-k^2)\}|b\_k|$$

It's known that when $p=2,3$, the conjecture is true for functions of the form $P\; \backslash circ\; \backslash phi$ where $P$ is a polynomial and $\backslash phi$ is univalent.

• {{cite journal |doi=10.1090/S0002-9947-1948-0023910-X|title=On some determinants related to 𝑝-valent functions |year=1948 |last1=Goodman |first1=A. W. |journal=Transactions of the American Mathematical Society |volume=63 |pages=175–192 |doi-access=free }}
• {{cite journal |doi=10.1090/S0002-9939-1978-0460619-7|title=Goodman's conjecture and the coefficients of univalent functions |year=1978 |last1=Lyzzaik |first1=Abdallah |last2=Styer |first2=David |journal=Proceedings of the American Mathematical Society |volume=69 |pages=111–114 |doi-access=free }}
• {{cite book |doi=10.1016/S1874-5709(02)80012-9 |chapter=Logarithmic Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains |title=Geometric Function Theory |series=Handbook of Complex Analysis |year=2002 |last1=Grinshpan |first1=Arcadii Z. |volume=1 |pages=273–332 |isbn=978-0-444-82845-3| url={{Google books|Wd7hKzGf9E8C|page=321|plainurl=yes}}}}
• {{cite journal |last1=AGrinshpan |first1=A.Z.|title=On the Goodman conjecture and related functions of several complex variables |journal=Department of Mathematics, University of South Florida, Tampa, FL |date=1997 |volume=9 |issue=3 |pages=198–204|url=http://m.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=788&what=fullt&option_lang=eng|mr=1466800}}
• {{cite journal |doi=10.1090/S0002-9939-1995-1242085-7|title=On an identity related to multivalent functions |year=1995 |last1=Grinshpan |first1=A. Z. |journal=Proceedings of the American Mathematical Society |volume=123 |issue=4 |page=1199 |doi-access=free }}

Category:Complex analysis

Category:Conjectures