Approximation of Certain Non-vanishing Analytic Functions in a Parabolic Region

In this work, we consider a class of analytic functions f defined in the unit disk for which the values of zf ' /f lie in a parabolic region of the right-half plane. By using a well-known sufficient condition for functions to be in this class in terms of the Taylor coefficients of z/f, we introduce a subclass F alpha of this class. The aim of the paper is to find the best approximation of non-vanishing analytic functions of the form z/f by functions z/g with g is an element of F alpha. The proof relies on solving a semi-infinite quadratic problem, a problem of independent interest.