Holomorphic functions on the disk with infinitely many zeros

Abstract

Suppose that $\varphi (z)={\sum }_{n=1}^{\infty }{d}_n{z}^n$ is a series with radius of convergence greater than 1 and suppose that $d_n$ are real numbers such that $\varphi (1)\ne 0$. We prove that for any integer $a$ greater than $1$, the complex variable function

has infinitely many zeros on the unit disk. It even takes every complex value in every disk centered in any point of the boundary.