The Krzyż Conjecture and an Entropy Conjecture

Abstract

We show that if the minimum entropy for a polynomial with roots on the unit circle is attained by polynomials with equally spaced roots, then, under a generic hypothesis about the nature of the extremum, the Krzyz conjecture on the maximum modulus of the Taylor coefficients of a holomorphic function that maps the disk to the punctured disk is true.

C*-Algebraic Bieberbach, Robertson, Lebedev-Milin, Zalcman, Krzyz and Corona Conjectures

“…In 1978 Horowitz proved that there exists 0< c < 1 such that |a n | ≤ c for all n [51]. In fact, c = 1 − 1 3πIn 2021, Agler and McCarthy obtained a connection between Conjecture 2.1 and the entropy conjecture for polynomials with zeros on the standard unit circle group[1]. An extension of Krzyz conjecture has been formulated by Samaris in 2001[101].We state C*-algebraic versions of Krzyz conjecture as follows.Conjecture 2.2.…”