Goodman's conjecture

Template:No footnotes Goodman's conjecture on the coefficients of multivalued functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.

Formulation

Let $f(z)=\sum _{n=1}^{\mathrm{\infty }}{b}_n{z}^n$ be a $p$-valent function. The conjecture claims the following coefficients hold: $$|b_n|\le \sum _{k=1}^p\frac{2k(n+p)!}{(p-k)!(p+k)!(n-p-1)!(n^2-k^2)}|b_k|$$

Partial results

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

External sources

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