The Krzy\.{z} conjecture revisited
Abstract
The Krzy\.z conjecture concerns the largest values of the Taylor coefficients of a non-vanishing analytic function bounded by one in modulus in the unit disk. It has been open since 1968 even though information on the structure of extremal functions is available. The purpose of this paper is to collect various conditions that the coefficients of an extremal function (and various other quantities associated with it) should satisfy if the conjecture is true and to show that each one of these properties is equivalent to the conjecture itself. This may provide several possible starting points for future attempts at solving the problem.
Cite
@article{arxiv.1311.7668,
title = {The Krzy\.{z} conjecture revisited},
author = {María J. Martín and Eric T. Sawyer and Ignacio Uriarte-Tuero and Dragan Vukotić},
journal= {arXiv preprint arXiv:1311.7668},
year = {2016}
}
Comments
Condition (d) in the main Theorem of version 1 of this paper was erroneous. This has been corrected in the subsequent versions v2 and v3. Also, a proof of one implication in the Main Theorem has now been made shorter with respect to v2