Generalization of certain well-known polynomial inequalities

If P(z)=∑ν=0ncνzν is a polynomial of degree n   having no zeros in |z|<1|z|<1, then for |β|⩽1|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|zP′(z)+nβ2P(z)|⩽n2{|1+β2|+|β2|}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.