Interpolation Sequences for the Class of Functions of Finite η-Type Analytic in the Unit Disk

Abstract

We establish conditions for the existence of a solution of the interpolation problem f(λ _n ) = b _n in the class of functions f analytic in the unit disk and such that

$$\left( {\exists \;c_1 > 0} \right)\;\left( {\forall z,\;|\;z\;| < 1} \right):\;\;\left| {f\left( z \right)} \right|\;\; \leqslant \;\;\;\exp \left( {c_1 \eta \left( {\frac{{c_1 }}{{1 - \left| z \right|}}} \right)} \right).$$

Here, η : [1; +∞) → (0; +∞) is an increasing function convex with respect to ln t on the interval [1; +∞) and such that ln t = o(η(t)), t → ∞.