On the derivative of a polynomial

Let P (z) = cnz + ∑n ν=μ cn−νz n−ν , 1 ≤ μ < n, be a polynomial of degree at most n having no zeros in |z| < k, k ≤ 1, and Q(z) = zP (1/z), it is proved by Dewan et al. [5] that if |P ′(z)| and |Q′(z)| becomes maximum at the same point on |z| = 1, then max |z|=1 |P (z)| ≤ n 1 + kn−μ+1 {max |z|=1 |P (z)| − min |z|=k |P (z)|}. In this paper, we generalize the above inequality for the polynomials of type P (z) = a0 + ∑n ν=μ aνz ν , 1 ≤ μ ≤ n.