A new inequality for a polynomial

Abstract

Let p(z)=a0+∑j=tnajzj be a polynomial of degree n, having no zeros in |z|<k, k≥1 then it has been shown that for R>1 and |z|=1, |p(Rz)−p(z)|≤(Rn−1)(1+AtBtKt+1)/(1+kt+1+AtBt(kt+1+k2t))max|z|=1|p(z)|−{1−(1+AtBtkt+1)/(1+kt+1+AtBt(kt+1+k2t))}((Rn−1)m/kn), where m=min|z|=k|p(z)|, 1≤t<n, At=(Rt−1)/(Rn−1), and Bt=|at/a0|. Our result generalizes and improves some well-known results.