The Journal of the Indian Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

Some Results Concerning Sendov Conjecture


Affiliations
1 Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir, India
     

   Subscribe/Renew Journal


Let P(z) be a complex polynomial of degree n having all its zeros in |z| ≤ 1. Then the Sendov’s Conjecture states that there is always a critical point of P(z) in |z − a| ≤ 1, where a is any zero of P(z). In this paper, we verify the Sendov’s Conjecture for some special cases. The case where a is the root of pth smallest modulus is also investigated.


Keywords

Polynomial, Disk, Zeros, Critical Point, Transformation.
Subscription Login to verify subscription
User
Notifications
Font Size


  • B. Bojanov, Q. Rahman and J. Szynal, On a conjecture of Sendov about the critical points of a polynomial, Mathematische Z., 190 (1985), 281–285.

  • J. E. Brown and G. Xiang, Proof of Sendov conjecture for polynomials of degree at most eight, J. Math. Anal. Appl., 2 (1999), 272–292.

  • G. L. Cohen and G. H. Smith, A Proof of Iliev’s conjecture for polynomials with three zeros, Amer. Math. Monthly, 95 (1988), 734–737.

  • W. K. Hayman, Research Problems in Function Theory, Althlone Press, London, 1967.

  • Q. I. Rahman and G. Schmeisser. Analytic Theory of Polynomials, Oxford University Press, 2002.

  • Z. Rubenstein, On a Problem of Ilye, Pacific J. Math., 26 (1968), 159–161.

  • Bl. Sendov, Generalization of a Conjecture in the geometry of plynomials, Serdica Math. J. 28 (2002), 283–304.

  • T. Sheil–Small, Complex Polynomials, Cambridge University Press, 2002.


Abstract Views: 231

PDF Views: 0




  • Some Results Concerning Sendov Conjecture

Abstract Views: 231  |  PDF Views: 0

Authors

Mohammad Ibrahim Mir
Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir, India
Ishfaq Nazir
Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir, India
Irfan Ahmad Wani
Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir, India

Abstract


Let P(z) be a complex polynomial of degree n having all its zeros in |z| ≤ 1. Then the Sendov’s Conjecture states that there is always a critical point of P(z) in |z − a| ≤ 1, where a is any zero of P(z). In this paper, we verify the Sendov’s Conjecture for some special cases. The case where a is the root of pth smallest modulus is also investigated.


Keywords


Polynomial, Disk, Zeros, Critical Point, Transformation.

References