On the zeros of solutions of an extremal problem in H 1

Abstract

For a nonzero function f in H 1, the classical Hardy space on the unit disc, we put .. The intersection of Sf and the unit sphere in H 1 is a solution set of a certain extremal problem in H 1. It is known that Sf can be represented in the form Sf = SB × g 0, where B is a Blaschke product and g 0 is a function in H 1 with Sg0 = {λ g0 : λ > 0}. Also it is known that the linear span of Sf is finite dimensional if and only if B is a finite Blaschke product, and when B is a finite Blaschke product, we can describe completely the set SB and the zeros of functions in SB . In this paper we study the set of zeros of functions in SB when B is an infinite Blaschke product whose set of singularities is not the whole circle. In particular, we study the behavior of zeros of functions in SB in the sectors of the form: δ= {reiθ:0 < r ≤ 1c1 < θ < c2 } on which the zeros of B has no accumulation points, and establish a convergence order theorem for the zeros in δ of functions in SB .