On sharpening of inequalities for a class of polynomials satisfying p(z)≡znp(1/z)

Abstract

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 [3], and Rahman and Tariq [12]. 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.