Krzyż's conjecture for the seventh coefficient

Abstract

A holomorphic function bounded in modulus by one in the unit disk has Taylor coefficients of modulus at most one. If the function has no zeros, Krzyż's conjecture predicts the stronger bound $2/e$ for every coefficient of positive order. We establish this bound for the seventh coefficient: $|a_7|\leq2/e$, with equality precisely for the functions $η\exp[-(1-ξz^7)/(1+ξz^7)]$, where $|η|=|ξ|=1$. Thus the largest possible value of $|f^{(7)}(0)|$ in this class is $7!\,2/e$, and the extremal functions are uniquely determined up to rotations of the variable and multiplication by a constant of modulus one.