Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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INEQUALITIES FOR THE DERIVATIVE OF POLYNOMIALS WITH RESTRICTED ZEROS

  • Rather, N.A. (Department of Mathematics, University of Kashmir) ;
  • Dar, Ishfaq (Department of Mathematics, University of Kashmir) ;
  • Iqbal, A. (Department of Mathematics, University of Kashmir)
  • Received : 2020.03.21
  • Accepted : 2020.12.09
  • Published : 2020.12.30
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Abstract

For a polynomial $P(z)={\sum_{{\nu}=0}^{n}}\;a_{\nu}z^{\nu}$ of degree n having all its zeros in |z| ≤ k, k ≥ 1, it was shown by Rather and Dar [13] that ${\max_{{\mid}z{\mid}=1}}{\mid}P^{\prime}(z){\mid}{\geq}{\frac{1}{1+k^n}}\(n+{\frac{k^n{\mid}a_n{\mid}-{\mid}a_0{\mid}}{k^n{\mid}a_n{\mid}+{\mid}a_0{\mid}}}\){\max_{{\mid}z{\mid}=1}}{\mid}P(z){\mid}$. In this paper, we shall obtain some sharp estimates, which not only refine the above inequality but also generalize some well known Turán-type inequalities.

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References

  1. Abdul Aziz, Inequalities for the derivative of a polynomial, Proc. Amer. Math. Soc. 89 (2) (1983), 259-266. https://doi.org/10.1090/S0002-9939-1983-0712634-5
  2. A. Aziz and Q. M. Dawood, Inequalities for a polynomial and its derivatives, J. Approx. Theory 54 (1998), 306-313. https://doi.org/10.1016/0021-9045(88)90006-8
  3. A. Aziz and N. A. Rather, Some Zygmund type Lq inequalities for polynomials, J. Math. Anal. Appl. 289 (2004), 14-29. https://doi.org/10.1016/S0022-247X(03)00530-4
  4. A. Aziz and N. A. Rather, Inequalities for the polar derivative of a polynomial with restricted zeros, Math. Bulk. 17 (2003), 15-28.
  5. A. Aziz and N. A. Rather, A refinement of a theorem of Paul Turan concerning polynomials, Math. Ineq. Appl. 1 (1998), 231-238. https://doi.org/10.7153/mia-2020-23-18
  6. C. Frappier, Q. I. Rahman and Rt. St. Ruscheweyh, New inequalities for polynomials, Trans, Amer. Math. Soc. 288 (1985),69-99. https://doi.org/10.2307/2000427
  7. V. N. Dubinin, Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros, J. Math. Sci. 143 (2007), 3069-3076. https://doi.org/10.1007/s10958-007-0192-4
  8. N. K. Govil, On the derivative of a polynomial, Proc. Amer. Math. Soc. 41 (1973), 543-546. https://doi.org/10.1090/S0002-9939-1973-0325932-8
  9. P. D. Lax, Proof of a conjecture of P. Erdos on the derivative of a poplynomial, Bull. Amer. Math. Soc.,50(1944), 509-513. https://doi.org/10.1090/S0002-9904-1944-08177-9
  10. G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World scientific Publishing Co., Singapore, (1994).
  11. R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128, 12 (2000), 3513-3517. https://doi.org/10.1090/S0002-9939-00-05463-0
  12. Q.I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, New York, 2002.
  13. N. A. Rather and Ishfaq Dar, Some applications of the Boundary Schwarz lemma for Polynomials with restricted zeros, Applied Mathematics E-Notes 20 (2020), 422-431.
  14. N. A. Rather, Ishfaq Dar and A. Iqbal, Some extensions of a theorem of Paul Turan concerning polynomials, Kragujevac Journal of Mathematics, 46 (6), 969-979.
  15. N. A. Rather, Ishfaq Dar and A. Iqbal, On a refinement of Turan's inequality, Complex Anal Synerg. 6 (21) (2020). https://doi.org/10.1007/s40627-020-00058-5.
  16. N. A. Rather and Suhail Gulzar, Generalization of an inequality involving maximum moduli of a polynomial and its polar derivative, Non linear Funct. Anal. and Appl. 19 (2014), 213-221.
  17. P. Turan, Uber die Ableitung von Polynomen, Compos. Math. 7 (1939), 89-95.