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

$${\displaystyle |b_{n}|\leq \sum _{k=1}^{p}{\frac {2k(n+p)!}{(p-k)!(p+k)!(n-p-1)!(n^{2}-k^{2})}}|b_{k}|}$$

Partial results

It's known that when ${\displaystyle p=2,3}$, the conjecture is true for functions of the form ${\displaystyle P\circ \phi }$ where ${\displaystyle P}$ is a polynomial and ${\displaystyle \phi }$ 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. Vol. 1. pp. 273–332. doi:10.1016/S1874-5709(02)80012-9. ISBN 978-0-444-82845-3.
• 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. MR 1466800.
• 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.