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Inequalities for a Polynomial and its Derivative

  • PUKHTA, MOHAMMAD SYED (Division of Agri. Statistics, Sher-e-Kashmir University of Agricultural Sciences and Technology of Kashmir)
  • Received : 2014.01.25
  • Accepted : 2014.08.21
  • Published : 2015.09.23

Abstract

In this paper we consider the class of polynomials of the type $p(z)=z^s(a_0+{\Sigma}_{j={\mu}}^{n-s}ajz^j)$, $1{\leq}{\mu}{\leq}n-s$, $0{\leq}s{\leq}n-1$ having some zeros at origin and rest of zeros on or outside the boundary of a prescribed disk, and obtain the generalization of well known results.

Keywords

Derivative;polynomial;inequality;Zeros

References

  1. M. S. Pukhta, Abdullah Mir and T. A. Raja, Note on a theorem of S. Bernstein, J. Comp. & Math. Sci., 1(14)(2010), 419-423.
  2. T. N. Chan and M. A. Malik, On Erdos-Lax Theorem, Proc. Indian Acad. Math. Sci., 92(3)(1983), 191-193. https://doi.org/10.1007/BF02876763
  3. K. K. Dewan and Sunil Hans, On extremal properties for the derivative of polynomials, Mathematica balkanica, 23(Fasc. 1-2)(2009), 27-35.
  4. S. Bernestein, Lecons sur less propries extremales et la meilleure dune functions rella, Paris, 1926.
  5. N. K. Govil, On a theorem of S. Bernstein, J. Math. and Phy. Sci., 14(1980), 183-187.
  6. P. D. Lax, Proof of a conjecture of P. Erdos on the derivative of a polynomial, Amer. Math. Soc. Bull., 50(1944), 509-513. https://doi.org/10.1090/S0002-9904-1944-08177-9
  7. M. A. Malik, On the derivative of a polynomial, J. London Math. Soc., 2(1)(1969).
  8. M. S. Pukhta, Extremal Problems for Polynomials and on location of zeros of polynomials, Ph.D. Thesis, Jamia Millia Islamia, New Delhi, (1995).
  9. M. A. Qazi, On the maximum modulus of polynomials, Proc. Amer. Math. Soc., 115(1992), 337-343. https://doi.org/10.1090/S0002-9939-1992-1113648-1
  10. N. K. Govil, Some inequalities for derivative of polynomials, J. Approx. Theory, 66(1)(1991), 29-35. https://doi.org/10.1016/0021-9045(91)90052-C