Nonlinear Functional Analysis and Applications

  • 제26권2호
  • /
  • Pages.373-380
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  • 2021
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  • 1229-1595(pISSN)
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  • 2466-0973(eISSN)
경남대학교 기초과학연구소 (The Basic Sciences Research Institute, Kyungnam University)
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ON AN INEQUALITY OF S. BERNSTEIN

  • 투고 : 2020.09.29
  • 심사 : 2021.02.09
  • 발행 : 2021.06.15
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초록

If $p(z)={\sum\limits_{{\nu}=0}^{n}}a_{\nu}z^{\nu}$ is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then Govil [3] proved that $${\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p^{\prime}(z){\mid}\;{\leq}\;{\frac{n}{k^n+k^{n-1}}}\;{\max\limits_{{\mid}z{\mid}=1}}\;{\mid}p(z){\mid}$$. In this paper, by involving certain coefficients of p(z), we not only improve the above inequality but also improve a result proved by Dewan and Mir [2].

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과제정보

The authors are extremely grateful to the referee for his valuable comments and suggestions about the paper.

참고문헌

  1. S. Bernstein, Lecons sur les proprietes extremales et la meilleure approximation desfonctions analytiques d'une variable reelle, Gauthier Villars, Paris, 1926.
  2. K.K. Dewan and A. Mir, Note on a theorem of S. Bernstein, Southeast Asian Bull. Math., 31 (2007), 691-695.
  3. N.K. Govil, On a Theorem of S. Bernstein, J. Math. Phy. Sci., 14(2) (1980), 183-187.
  4. N.K. Govil, On a Theorem of S. Bernstein, Proc. Nat. Acad. Sci., 50 (1980), 50-52.
  5. N.K. Govil and Q.I. Rahman, Functions of exponential type not vanishing in half plane and related polynomials, Trans. Amer. Math. Soc., 137 (1969), 501-517. https://doi.org/10.1090/S0002-9947-1969-0236385-6
  6. N.K. Govil, Q. I. Rahman and G. Schmeisser, On the derivative of a polynomial, Illinois J. Math., 23(2) (1979), 319-329. https://doi.org/10.1215/ijm/1256048243
  7. M.A. Malik, On the derivative of a polynomial, J. London Math. Soc., 1 (1969), 57-60. https://doi.org/10.1112/jlms/s2-1.1.57
  8. P.D. Lax, Proof of a Conjecture of P. Erdos on the derivative of a polynomial, Bull. Amer. Math. Soc., 50 (1944), 509-513. https://doi.org/10.1090/S0002-9904-1944-08177-9