On the critical points of entire functions

TL;DR

This paper extends known results about the location of critical points of polynomials to certain classes of entire functions, showing under specific conditions that their derivatives have no roots in particular regions.

Contribution

It generalizes previous polynomial results to entire functions, establishing conditions on zeros that prevent critical points from appearing in certain areas.

Findings

Critical points of entire functions are absent in specific regions under certain zero conditions.

The results extend Sendov's conjecture-related findings from polynomials to entire functions.

Provides bounds on regions where derivatives of entire functions have no zeros.

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 $|a_n|$.