Parametrical Representations of Some Classes of Holomorphic Functions in the Disk

Abstract

Let D = {z : |z| < 1} be the unit disk on the complex plane C, let Γ be its boundary, and let H(D) be set of all holomorphic functions in D. For α > 0, we denote by N _α the following class of functions:

$$N_\alpha = \left\{ {f \in H\left( D \right):T\left( {r,f} \right) \leqslant \frac{{c_f }} {{\left( {1 - r} \right)^\alpha }};0 \leqslant r \leqslant 1} \right\},$$

where T(r, f) is the Nevanlinna characteristic of f. It is obvious that if α = 0, then N ₀ = N (N is the standard Nevanlinna class). The following classical result of R. Nevanlinna (see [1]) is well known: f ∈ N if and only if

$$f\left( z \right) = c_\lambda z^\lambda B\left( {z,z_k } \right)\exp \int_{ - \pi }^\pi {\frac{{d\mu \left( \theta \right)}} {{\left( {1 - e^{ - i\theta } z} \right)}}} , z \in D,$$

where c _λ is a complex number, λ is a non-negative integer, B(z, z _k ) is a Blaschke product,{z _k } ^+∞ _k=1 is any sequence of points in D satisfying

$$\sum\limits_{k = 1}^{ + \infty } {\left( {1 - \left| {z_k } \right|} \right) < + \infty ,}$$

μ(θ) is any real measure on [−π,π].