Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지
DOI QR코드

DOI QR Code

ON AN OPERATOR PRESERVING POLYNOMIAL INEQUALITIES

  • Rather, N.A. (Department of Mathematics, University of Kashmir) ;
  • Ali, Liyaqat (Department of Mathematics, University of Kashmir) ;
  • Dar, Ishfaq (Department of Mathematics, University of Kashmir)
  • Received : 2021.04.07
  • Accepted : 2021.06.11
  • Published : 2021.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider an operator N : 𝓟n → 𝓟n on the space of polynomials 𝓟n of degree at most n and establish some compact generalizations of Bernstein-type polynomial inequalities, which include several well known polynomial inequalities as special cases.

Keywords

References

  1. N.C. Ankeny and T.J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math. 5 (1995), 849-852. https://doi.org/10.2140/pjm.1955.5.849
  2. A. Aziz and N. A. Rather, Some compact generalizations of Bernstien-type inequalities for polynomials, Math. Inequal. Appl. 7(3) (2004), 393-403.
  3. S. N. Bernstein, Sur. lordre de la meilleure appromation des functions continues par des Polynomes de degree donne, Mem. Acad. Roy. Belgique 12(4), (1912) 1-103.
  4. S.N. Bernstein, On the best approximation of continuous functions by polynomials of given degree (O nailuchshem problizhenii nepreryvnykh funktsii posredstrvom mnogochlenov dannoi stepeni). Sobraniye sochinenii 1, 11-104 (1952). (1912, Izd. Akad. Nauk SSSR, Vol. I).
  5. J.P. Kahane, Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985).
  6. G.G. Lorentz, Approximation of Functions, 2nd edn. Chelsea Publishing Co. New York (1986).
  7. M. Marden, Geometry of Polynomials, Math. Surveys, No. 3, Amer. Math. Soc. Providence, RI, 1966.
  8. G. V. Milovanovic, D. S. Mitrinovic and Th. M. Rassias, Topics In Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publishing Co. Singapore, (1994).
  9. P. J. O'hara and R. S. Rodriguez, Some properties of self-inversive polynomials, Proc. Amer. Math. Soc. 44 (1974), 331-335. https://doi.org/10.1090/S0002-9939-1974-0349967-5
  10. G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, 94. Cambridge University Press, Cambridge (1989).
  11. G. Polya and G. Szego, Aufgaben und lehrsatze aus der Analysis, Springer-Verlag, Berlin, 1925.
  12. N. A. Rather, Ishfaq Dar, Suhail Gulzar, On the zeros of certain composite polynomials and an operator preserving inequalities, The Ramanujan Journal doi.org/10.1007/s11139-020-00261-2.
  13. H. Queffelec, M. Queffelec, Diophantine approximation and Dirichlet series. In: HRI Lecture Notes Series, 2 (2013).
  14. Q. I. Rahman , G. Schmeisser, Analytic Theory of Polynomials, Clarendon Press Oxford (2002), 243-270.
  15. R. Salem, A. Zygmund, Some properties of trigonometric series whose terms have random sign. Acta. Math. 91, 245-301 (1954). https://doi.org/10.1007/BF02393433