Sharp coefficient-difference bound for normalized univalent functions

Determine the best possible upper bound for the difference of successive coefficient moduli, d_n=\left||a_{n+1}|-|a_n|\right|, for every normalized univalent function f(z)=z+\sum_{n=2}^{\infty}a_nz^n in the unit disk.

Background

The paper introduces the coefficient-difference quantity d_n=\left||a_{n+1}|-|a_n|\right| for normalized univalent functions in the unit disk. It reports that Hayman established the existence of an absolute constant A with d_n\leq A, while subsequent work successively improved numerical upper bounds to values below 9, 4.26, and 3.61.

The sharp bound is known in some restricted settings: for n=2, a sharp inequality is available for the full class of normalized univalent functions, and for starlike functions the best possible uniform bound is A=1. However, the paper explicitly identifies the optimal bound for general normalized univalent functions as unresolved.