Argument of bounded analytic functions and Frostman’s type conditions

Abstract

We describe the growth of the naturally defined argument of a bounded analytic function in the unit disk in terms of the complete measure introduced by A.Grishin. As a consequence, we characterize the local behavior of a logarithm of an analytic function. We also find necessary and sufficient conditions for closeness of logf(z), f 2 H 1 , and the local concentration of the zeros of f. One of the basic theorems in complex analysis is the Argument principle, which states that if f(z) is a meromorphic function inside and on some closed contour γ, with f having no zeros or poles on γ, then the increase of Arg f(z) along γ divided over 2π is equal to N −P, where N and P denote respectively the number of zeros and poles of f(z) inside the contour γ. It seems reason- able to ask what can be said if the number of zeros (poles) of f is infinite. Obviously, the contour should contain a singular point and the increase of Arg f(z) along γ need not be bounded in this case. Theorem 2 of this paper can be considered as a generalization of the Argument principle for bounded analytic functions in the unit disk D = {z ∈ C : |z| 0, can be represented in the form