Outer Functions in yet Another Class of Analytic Functions

Abstract

Let ω be a positive continuous function on (0, π] satisfying the condition

$$\begin{array}{*{20}{c}} {{{c}_{1}}{{{\left( {\frac{x}{y}} \right)}}^{\alpha }} \leqslant \frac{{\omega (y)}}{{\omega (x)}} \leqslant {{c}_{2}}{{{\left( {\frac{y}{x}} \right)}}^{\alpha }},} & {0 < x} \\ \end{array} < y \leqslant \pi ,$$

with some 0 < α < 1. For a natural number n, let Λⁿ⁻¹ Z _ω , denote the class of functions f analytic in the unit disc D and continuous in $\bar D$ such that the derivatives f′, ..., f ⁽ⁿ⁻¹⁾ possess the same property and, moreover,

$$\begin{gathered} \left| {f^{\left( {n - 1} \right)} \left( {e^{i\left( {\theta + h} \right)} } \right) - 2f^{\left( {n - 1} \right)} \left( {e^{i\theta } } \right) + f^{\left( {n - 1} \right)} \left( {e^{i\left( {\theta - h} \right)} } \right)} \right| \leqslant c_f h \omega \left( h \right), \hfill \\ 0 < h \leqslant \pi , 0 \leqslant \theta < \pi . \hfill \\ \end{gathered}$$

The outer functions (in the sense of the Nevanlinna inner-outer factorization) belonging to Λⁿ⁻¹ Z _ω , are completely described.