For a polynomial p(z) of degree n, we consider an operator Dα which map a polynomial p(z) into Dαp(z): = (α − z)p′(z) + np(z) with respect to α. It was proved by Liman et al. (2010) that if p(z) has no zeros in |z | < 1, then for all α, β ∈ ℂ with |α | ≥ 1, | β | ≤ 1 and |z | = 1, |zDαp(z) + nβ((|α | − 1)/2)p(z)|≤(n/2){[|α + β((|α | − 1)/2)|+|z + β((|α | − 1)/2)|]max|z|=1 | p(z)|−[|α + β((|α | − 1)/2)|−|z + β((|α | − 1)/2)|]min|z|=1 | p(z)|}. In this paper we extend the above inequality for the polynomials having no zeros in |z | < k, where k ≤ 1. Our result generalizes certain well‐known polynomial inequalities.