Inequalities for the derivative of a polynomial

Let P(z) be a polynomial of degreen which does not vanish in ¦z¦ <k, wherek > 0. Fork ≤ 1, it is known that

$$mathop {max }limits_{|z| = 1} |P'(z)| leqslant frac{n}{{1 + k^n }}mathop {max }limits_{|z| = 1} |P(z)|$$

, provided ¦P’(z)¦ and ¦Q’(z)¦ become maximum at the same point on ¦z¦ = 1, where(Q(z) = z^n overline {P(1/bar z)} ). In this paper we obtain certain refinements of this result. We also present a refinement of a generalization of the theorem of Tu?an.