Generalization of certain well-known polynomial inequalities

Abstract

If P ( z ) = ∑ ν = 0 n c ν z ν is a polynomial of degree n having no zeros in | z | < 1 , then for | β | ⩽ 1 , it was proved by Jain [V.K. Jain, Generalization of certain well known inequalities for polynomials, Glas. Mat. 32 (52) (1997) 45–51] that | z P ′ ( z ) + n β 2 P ( z ) | ⩽ n 2 { | 1 + β 2 | + | β 2 | } max | z | = 1 | P ( z ) | , | z | = 1 . In this paper, we shall first obtain a result concerning minimum modulus of polynomials and next we improve upon the above inequality for the polynomials with restricted zeros. Our results refine and generalize certain well-known polynomial inequalities including some results of Bernstein, Lax, Malik and Vong, and Aziz and Dawood.