On the lemniscate components containing no critical points of a polynomial except for its zeros

TL;DR

This paper establishes a precise inequality relating to the behavior of a polynomial's logarithmic derivative within certain lemniscate regions that lack critical points, advancing understanding of polynomial dynamics.

Contribution

It introduces a sharp inequality for the polynomial's logarithmic derivative in lemniscate components free of critical points, a novel result in complex analysis.

Findings

Proved a sharp inequality for the logarithmic derivative in specific lemniscate regions.

Identified conditions under which lemniscate components contain no critical points.

Enhanced understanding of polynomial behavior in complex regions.

Abstract

We prove a sharp inequality for the modulus of the logarithmic derivative of a polynomial in the lemniscate components containing no critical points.