On the number of zeros of a polynomial in a specified disk

TL;DR

This paper investigates how restrictions on the coefficients of a polynomial's real part influence the maximum number of zeros it can have within a specific disk, providing bounds based on these constraints.

Contribution

It introduces new coefficient restrictions on the real part of polynomials to estimate the maximum number of zeros in a given disk.

Findings

Derived bounds on zeros based on coefficient restrictions

Extended classical zero distribution results

Provided estimates for zeros in specified regions

Abstract

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ may be complex. We impose some restriction on the coefficients of the real part of the given polynomial and then estimate the maximum number of zeros such polynomial can possibly have in a specified disk.