Bernstein-type inequalities for N-Operator

Abstract

In this paper, we investigate various comparison inequalities for a certain class of linear differential
operators acting on complex polynomials. We introduce a generalized operator, called the N-operator,
which encompasses extensions of several polynomial inequalities and specializes to the classical B-operator
under suitable parameter selections. Assuming natural constraints on the zeros of the involved polynomials,
we establish new Bernstein-type inequalities in the uniform norm that compare the action of
N on two polynomials, potentially of different degrees. Our results generalize and sharpen classical inequalities—
such as those of Erd˝os–Lax and Ankeny–Rivlin—while also relaxing the assumptions made in
earlier works, including those by Rather, Gulzar, Mir and others. Several known results are recovered as
special cases, and our framework highlights the analytic behavior of polynomials under zero-preserving
differential operators.
1. Introduction and Preliminaries
The investigation of comparison inequalities