Some properties of extremal functions in the Krzyz problem

Abstract

Krzyz's conjecture on estimating Taylor coefficient moduli in the class of ho\-lo\-mor\-phic, bounded, nonvanishing functions is considered. Unlike many studies of this problem, this paper studies global extremals as well as locally extremal functions in the introduced topology, which is stronger than the topology of locally uniform convergence. The relation between the class under consideration and the Caratheodory class is established, and the general form of extremal functions in the Krzyz problem is described. It is shown that every extremal function determines through its coefficients a polynomial $H$ with positive real part in the unit disk. A finite-dimensional analogue of the Caratheodory-Toeplitz criterion for polynomials, following from the Fejer-Riesz theorem, is used to study such polynomials. Necessary extremality conditions are obtained that substantially simplify the study of extremals for fixed $n$. Conditions for the uniqueness of an extremal function are investigated. In particular, it is proved that uniqueness of a global extremal implies the Krzyz conjecture. The notion of functions of extremal type, which satisfy all the necessary extremality conditions obtained, is introduced. Every locally extremal function is of extremal type, and the search for such functions at a fixed coefficient index reduces to a finite system of equations and inequalities. The sets of functions of extremal type are completely described for $n=1,2,3$. It is proved that these sets contain one function for $n=1$, five functions for $n=2$, and nineteen functions for $n=3$. Comparing the values of the functional under study on these sets proves the Krzyz conjecture for $n=1,2,3$.