SOME POLYNOMIAL INEQUALITIES IN THE COMPLEX DOMAIN

摘要

Let P(z)=∑nν=0cνzνbe a polynomial of degree n and let M(f,r) = max|z|=r |f(z)| for an arbitrary entire functionf (z). If P(z) has no zeros in |z| < 1 with M(P, 1) = 1, then for |α|≤ 1, it is proved by Jain[Glasnik Matematiki, 32(52) (1997), 45-51] that |P(Rz) + α(R+1/2)nP(z)|≤1/2{|1+α(R+1/2)n|+|Rn+α(R+1/2)n|},R≥1,|z|=1.In this paper, we shall first obtain a result concerning minimum modulus of polynomials and next improve the above inequality for polynomials with restricted zeros. Our result improves the well known inequality due to Ankeny and Rivlin[1] and besides generalizes some well known polynomial inequalities proved by Aziz and Dawood[3].