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  • 1 Department of Mathematics, Birla Institute of Technology and Science Pilani, K K Birla GoaCampus, Goa 403726, India
  • 2 e-mail: p20180024@goa.bits-pilani.ac.in
  • 3 e-mail: prasannak@goa.bits-pilani.ac.in
  • 4 Department of Mathematics and Statistics, Auburn University, Auburn, U.S.A
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Abstract

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) ≡ 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.

  • [1]

    Aziz, A., Inequalities for the derivatives of a polynomial, Proc. Amer. Math. Soc., 89 (1983), 259266.

  • [2]

    Bernstein, S. N., Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle, Gauthier-Villars, Paris, 1926.

    • Search Google Scholar
    • Export Citation
  • [3]

    Datt, B. and Govil, N. K., Some inequalities for polynomials satisfying p(z) ≡ z n p(1/z), Approx. Theory Appl ., 12 (1996), 4044.

  • [4]

    Frappier, C. and Rahman, Q. I., On an inequality of S. Bernstein, Can. J. Math., 34 (1982), 932944.

  • [5]

    Frappier, C., Rahman, Q. I. and Ruscheweyh, St., New inequalities for polynomials, Trans. Amer. Math. Soc., 288 (1985), 6999.

  • [6]

    Govil, N. K., On the derivative of a polynomial, Proc. Amer. Math. Soc., 41 (1973), 543546.

  • [7]

    Govil, N. K., Jain, V. K. and Labelle, G., Inequalities for polynomials satisfying p(z) ≡ z n p(1/z), Proc. Amer. Math. Soc., 57 (1976), 238242.

    • Search Google Scholar
    • Export Citation
  • [8]

    Govil, N. K. and Vetterlein, D. H., Inequalities for a class of polynomials satisfying p(z) ≡ z n p(1/z), Complex Variables, 31 (1996), 185191.

    • Search Google Scholar
    • Export Citation
  • [9]

    Jain, V. K., Inequalities for polynomials satisfying p(z) ≡ z n p(1/z) II, J. Indian Math. Soc., 59 (1993), 167170.

  • [10]

    Milovanović, G. V. , Mitrinović, D. S. and Rassias, Th. M., Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World scientific Publishing Co. Singapore, 1994.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [11]

    Rahman, Q. I. and Schmeisser, G., Analytic Theory of Polynomials, Oxford University Press, 2002.

  • [12]

    Rahman, Q. I. and Tariq, Q. M., An inequality for ‘self-reciprocal’ polynomials, East J. Approx., 12 (2006), 4351.

  • [13]

    Shaeffer, A. C., Inequalities of A. Markov and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc., 47 (1941), 567579.

    • Search Google Scholar
    • Export Citation