A new sharp inequality for arbitrary complex polynomials related to Gamma-lines

ABSTRACT

Let P(z) be an arbitrary complex polynomial and $\mathrm{\Gamma }$ an arbitrary straight line in the complex plane $\mathbb{C}$. We study the curves $\chi$ in a given domain D such that $P(z)\in \mathrm{\Gamma }$ for all $z\in \chi$. A simple sharp estimate for the length of these curves is obtained.

KEYWORDS:

• Complex polynomials
• Gamma-lines
• level sets

AMS SUBJECT CLASSIFICATIONS:

• 30C10
• 11C08

Notes

No potential conflict of interest was reported by the authors.