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

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 ) − ρ ) , 0 < r < 1 , ρ ∈ ( 0 , + ∞ ) , 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.