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Goodman's conjecture

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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 ) = ∑ n = 1 ∞ b n z n {\displaystyle f(z)=\sum _{n=1}^{\infty }{b_{n}z^{n}}} be a p {\displaystyle p} -valent function. The conjecture claims the following coefficients hold: | b n | ≤ ∑ k = 1 p 2 k ( n + p ) ! ( p − k ) ! ( p + k ) ! ( n − p − 1 ) ! ( n 2 − k 2 ) | b k | {\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 p = 2 , 3 {\displaystyle p=2,3} , the conjecture is true for functions of the form P ∘ ϕ {\displaystyle P\circ \phi } where P {\displaystyle P} is a polynomial and ϕ {\displaystyle \phi } is univalent.

External sources

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