Goodman's conjecture

Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.

Formulation

Let [math]\displaystyle{ f(z)= \sum_{n=1}^{\infty}{b_n z^n} }[/math] be a [math]\displaystyle{ p }[/math]-valent function. The conjecture claims the following coefficients hold: [math]\displaystyle{ |b_n| \le \sum_{k=1}^{p} \frac{2k(n+p)!}{(p-k)!(p+k)!(n-p-1)!(n^2-k^2)}|b_k| }[/math]

Partial results

It's known that when [math]\displaystyle{ p=2,3 }[/math], the conjecture is true for functions of the form [math]\displaystyle{ P \circ \phi }[/math] where [math]\displaystyle{ P }[/math] is a polynomial and [math]\displaystyle{ \phi }[/math] is univalent.

External sources

• Goodman, A. W. (1948). "On some determinants related to 𝑝-valent functions". Transactions of the American Mathematical Society 63: 175–192. doi:10.1090/S0002-9947-1948-0023910-X. 
• Lyzzaik, Abdallah; Styer, David (1978). "Goodman's conjecture and the coefficients of univalent functions". Proceedings of the American Mathematical Society 69: 111–114. doi:10.1090/S0002-9939-1978-0460619-7. 
• Grinshpan, Arcadii Z. (2002). "Logarithmic Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains". Geometric Function Theory. Handbook of Complex Analysis. 1. pp. 273–332. doi:10.1016/S1874-5709(02)80012-9. ISBN 978-0-444-82845-3. https://books.google.com/books?id=Wd7hKzGf9E8C&pg=PA321. 
• AGrinshpan, A.Z. (1997). "On the Goodman conjecture and related functions of several complex variables". Department of Mathematics, University of South Florida, Tampa, FL 9 (3): 198–204. http://m.mathnet.ru/php/getFT.phtml?jrnid=aa&paperid=788&what=fullt&option_lang=eng. 
• Grinshpan, A. Z. (1995). "On an identity related to multivalent functions". Proceedings of the American Mathematical Society 123 (4): 1199. doi:10.1090/S0002-9939-1995-1242085-7.