Interpolation of analytic functions of moderate growth in the unit disc and zeros of solutions of a linear differential equation

Abstract

In 2002 A.\ Hartmann and X.\ Massaneda obtained necessary and sufficient conditions for interpolation sequences for classes of analytic functions in the unit disc such that $\log M(r,f)=O((1-r)^{-\rho})$, $0<r<1$, $\rho \in (0 , +\infty)$, where $M(r,f)=\max\{ |f(z)|: |z|=r\}$. Using another method, we give an explicit construction of an interpolating function in this result. As an application we describe minimal growth of the coefficient $a$ such that the equation $f''+a(z)f=0$ possesses a solution with a prescribed sequence of zeros.