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Vol. 50 No. 2 (2024)
Serdica Mathematical Journal

Articles

On the complex polynomial inequalities concerning some operators

https://doi.org/10.55630/serdica.2024.50.151-172

Published 13-08-2024

  • Tawheeda Akhter
    • ,
  • Shabir A. Malik

Tawheeda Akhter
Lovely Professional University

Shabir A. Malik
University of Kashmir

Abstract

In this paper, we establish some new polynomial inequalities involving various operators and their relationship with the existing results. All these inequalities have been proved under the assumption that the underlying polynomial is being constrained with respect to its zeros.

2020 Mathematics Subject Classification:

30A10, 30C10, 30C15

Keywords

polynomial, zeros, inequalities in complex plane, Rouche's theorem, B-operator, polar derivative

Full text

Author Details

Tawheeda Akhter

Department of Mathematics
Lovely Professional University
Punjab, India
e-mail: takhter595@gmail.com

Shabir A. Malik

Department of Mathematics
University of Kashmir
Srinagar 190006, India
e-mail: shabirams2@gmail.com

References

  1. A. Aziz, Q. M. Dawood. Inequalities for a polynomial and its derivative. J. Approx. Theory 54, 3 (1988), 306–313.
  2. A. Aziz, W. Mohammad Shah. Some inequalities for the polar derivative of a polynomial. Proc. Indian Acad. Sci. Math. Sci. 107, 3 (1997), 263–270.
  3. S. Bernstein. Sur la limitation des dérivées des polynomes. C. R. Acad. Sci. Paris, 190, (1930), 338–341.
  4. M. Bidkham, H. A. Soleiman Mezerji. An operator preserving inequalities between polar derivative of a polynomial. J. Interdiscip. Math. 14, 5–6 (2011), 591–601, https://doi.org/10.1080/09720502.2011.10700776.
  5. P. Erdös. On extremal properties of derivatives of polynomials. Ann. of Math. (2) 41 (1940), 310–313.
  6. V. K. Jain. Generalization of an inequality involving maximum moduli of a polynomial and its polar derivative. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 50(98), 1 (2007), 67–74.
  7. P. D. Lax. Proof of a conjecture of P. Erd¨os on the derivative of a polynomial. Bull. Amer. Math. Soc. 50 (1944), 509–513.
  8. A. Liman, R. N. Mohapatra, W. M. Shah. Inequalities for the polar derivative of a polynomial. Complex Anal. Oper. Theory 6, 6 (2012), 1199–1209.
  9. M. A. Malik, M. C. Vong. Inequalities concerning the derivative of polynomials. Rend. Circ. Mat. Palermo (2) 34, 3 (1985), 422–426.
  10. S. A. Malik, B. A. Zargar. On a Generalization of an operator preserving Turán-type inequality for complex polynomials. Izv. Nats. Akad. Nauk Armenii Mat. 58, 5 (2023), 49–55; English translation in: J. Contemp. Math. Anal. 58, 5 (2023), 341–346.
  11. S. A. Malik, B. A. Zargar. On some operator preserving inequalities between polynomials. Asian-Eur. J. Math. 16, 11 (2023), Paper No. 2350197, 9 pp, https://doi.org/10.1142/S1793557123501978.
  12. M. Marden. Geometry of polynomials, 2nd Edition. Mathematical Surveys No. 3. Providence R.I., Amer. Math. Soc., 1966.
  13. Q. I. Rahman. Functions of exponential type. Trans. Amer. Math. Soc. 135 (1969), 295–309.
  14. Q. I. Rahman, G. Schmeisser. Analytic theory of polynomials. London Math. Soc. Monogr. (N.S.), vol. 26. Oxford, The Clarendon Press, Oxford University Press, 2002.
  15. W. Mohammad Shah. A generalization of a theorem of Paul Turán. J. Ramanujan Math. Soc. 11, 1 (1996), 67–72.