Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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LP-TYPE INEQUALITIES FOR DERIVATIVE OF A POLYNOMIAL

  • Received : 2021.07.04
  • Accepted : 2021.12.07
  • Published : 2021.12.30
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Abstract

For the polynomial P(z) of degree n and having all its zeros in |z| ≤ k, k ≥ 1, Jain [6] proved that $${{\max\limits_{{\mid}z{\mid}=1}}\;{\mid}P^{\prime}(z){\mid}{\geq}n\;{\frac{{\mid}c_0{\mid}+{\mid}c_n{\mid}k^{n+1}}{{\mid}c_0{\mid}(1+k^{n+1})+{\mid}c_n{\mid}(k^{n+1}+k^{2n})}\;{\max\limits_{{\mid}z{\mid}=1}}\;{\mid}P(z){\mid}$$. In this paper, we extend above inequality to its integral analogous and there by obtain more results which extended the already proved results to integral analogous.

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References

  1. A. Aziz, Integral mean estimates for polynomials with restricted zeros, J. Approx. 55 (1988), 232-238. https://doi.org/10.1016/0021-9045(88)90089-5
  2. S. Bernstein, Sur'e ordre de la meilleure approximation des functions continues par des polynomes de degr'e donn'e, Mem. Acad. R. Belg., 4 (1912), 1-103.
  3. 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
  4. N. K. Govil, Some inequalities for derivatives of polynomials, J. Approx. Theory, 66 (1991), 29-35. https://doi.org/10.1016/0021-9045(91)90052-c
  5. G. H. Hardy, The mean value of the modulus of an analytic function, Proc. London Math. Soc., 14 (1915), 319-330.
  6. V. K. Jain,On the derivative of a polynomial, Bull. Math. Soc. Sci. Math. Roumanie Tome, 59 (2016), 339-347.
  7. M.A. Malik, On the derivative of a polynomial; j. Lond. Math. Soc. 1 (1969), 57-60. https://doi.org/10.1112/jlms/s2-1.1.57
  8. M. A. Malik, An integral mean estimate for polynomials, Proc. Amer. Math. Soc., 91 (1984), 281-284. https://doi.org/10.1090/S0002-9939-1984-0740186-3
  9. G. V. Milovanovic, D. S. Mitrinovic and T. M. Rassias, Topics in Polynomials, Extremal problems, Inequalities, Zeros , World Scientific, Singapore, (1994).
  10. Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, (2002).
  11. P. Turan, Uber die Ableitung von Polynomen Compos. Math. 7 , 89-95(1939)
  12. E. C. Titchmarsh, The theory of functions, The English Book Society and Oxford University Press, London. 1962.