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 $$f(z)= \sum_{n=1}^{\infty}{b_n z^n}$$ 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 $$P \circ \phi$$ where $$P$$ is a polynomial and $$\phi$$ is univalent.