No abstract

No abstract

We obtain a description of zeros, singular boundary functions, and modules of angular boundary values of the functions f 0 that are analytic in the half-plane C + = { z : Re z > 0 } and satisfy the conditionwhere 0 ≤ σ < + ∞ is a given number and η is a positive function continuously differentiable on [ 0; + ∞ ) and such that t t tLet A η ( σ ) denote the class of functions f 0 analytic in C + = { z : Re z > 0 } and satisfying the conditionwhere 0 ≤ σ < + ∞ is a given number.In [1], the zeros and, in [2], the singular boundary functions from the class A η ( σ ) are described. In [3, 4], necessary and sufficient conditions are established for an analytic function in C + to have a finite formal type if the improved order is given. In this paper, we obtain a parametric representation of the class A η ( σ ) up to the multiplier exp ( c z ), and describe the class of complete measures (in the terminology of Grishin [4]) of functions from the class A η ( σ ). Theorem 1 below, which may be regarded as an analog of a statement well known for the functions analytic and bounded in C + , implies, in particular, the above-mentioned results of [1, 2]. Theorem 1. For the existence of a function f ∈ A η ( σ ) with the given sequence of zeros ( λ n ) , λ n ∈ C + , the singular boundary function h : R → R , and the moduli of angular boundary values equal to | f ( i t ) |

No abstract

No abstract