Let be a polynomial of degree n. Further, let and . Then according to the well-known Bernstein inequalities, we have and . It is an open problem to obtain inequalities analogous to these inequalities for the class of polynomials satisfying p(z) ≡ znp(1/z). In this paper we obtain some inequalites in this direction for polynomials that belong to this class and have all their coefficients in any sector of opening γ, where 0 γ < π. Our results generalize and sharpen several of the known results in this direction, including those of Govil and Vetterlein , and Rahman and Tariq . We also present two examples to show that in some cases the bounds obtained by our results can be considerably sharper than the known bounds.
Aziz, A., Inequalities for the derivatives of a polynomial, Proc. Amer. Math. Soc., 89 (1983), 259–266.