Bernstein Type Inequalities for Composite Polynomials

Abstract

Establishing the lower and upper bound estimates for the maximum modulus of the derivative of composition of polynomials p(q(z)), where q(z) is a polynomial of degree m is an intriguing problem in geometric theory of polynomials.
In this paper, the maximum modulus for composite polynomials of Bernstein type is taken up with constraints such as the given polynomial does not vanish in the disc |z| < k, where k ≥ 1 which in particular yields some known inequalities of this type as special cases. In addition, the case when all the zeros of the underlying polynomial lie in |z| ≤ k, where k ≤ 1 is also considered.

Key words and phrases. Bernstein’s inequality, Polynomial, Zeros, Composition of polynomials.

2010 Mathematics Subject Classification. 30C10, 30A10, 30C15, 30E10