On the critical points of entire functions

Abstract

Several years ago, Aziz and Zargar, while considering some questions related to Sendov's conjecture, solved a problem posed by Brown (see [1,2]), showing that any complex polynomial of degree $n$ with a single zero at $z=0$ does not have any critical point in $B(0,1/n)$. More recently, this result has been generalised in [3] by Zargar and Ahmad. The aim of our paper is to extend the result to some classes of complex entire functions. We will show that, under some conditions on the zeros $a_n$ of $f$, $f'$ has no roots in $B(0,t) \setminus \lbrace 0 \rbrace$ for a certain $t$ depending on the values of $|a_n|$.