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  • Author or Editor: P. Kumar x
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Let p(z)=j=0najzj be a polynomial of degree n. Further, letM(p,R)=max|z|=R1|p(z)|, and p=M(p,1). Then according to the well-known Bernstein inequalities, we have pnp and M(p,R)Rnp. It is an open problem to obtain inequalities analogous to these inequalities for the class of polynomials satisfying p(z) ≡ z n p(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.

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