Structural aspects of extremal functions in the Krzy\.z conjecture

TL;DR

This paper advances understanding of extremal functions related to the Krzyż conjecture by establishing bounds, new formulas, and conditions that could lead to a proof of the conjecture.

Contribution

It provides a lower bound on the number of atoms in extremal functions, new formulas via variational methods, and characterizations of invariants linked to the Krzyż conjecture.

Findings

Lower bound on atoms: N ≥ c·n

New formulas for extremal functions

Conditions equivalent to the Krzyż conjecture

Abstract

Extremal functions for the $n$th coefficient in the Krzy\.z conjecture are atomic singular inner functions with at most $n$ atoms. This paper gives a lower bound on the number of atoms $N$ of the form $N\geq cn$, marking progress toward proving the expected $N=n$. Furthermore, we prove new formulas for extremal functions using variational techniques. Using these results and several other methods, we establish new conditions on extremal functions which are equivalent to the Krzy\.z conjecture being true. We also characterize the possible analytic invariants of extremal functions.