Some Results Concerning Sendov Conjecture

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Authors

  • Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir ,IN
  • Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir ,IN
  • Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101, Jammu and Kashmir ,IN

DOI:

https://doi.org/10.18311/jims/2023/28314

Keywords:

Polynomial, Disk, Zeros, Critical Point, Transformation.

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.

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Published

2023-03-24

How to Cite

Mir, M. I., Nazir, I., & Wani, I. A. (2023). Some Results Concerning Sendov Conjecture. The Journal of the Indian Mathematical Society, 90(1-2), 159–164. https://doi.org/10.18311/jims/2023/28314
Received 2021-07-27
Accepted 2022-04-15
Published 2023-03-24

 

References

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. DOI: https://doi.org/10.1007/BF01160464

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. DOI: https://doi.org/10.1006/jmaa.1999.6267

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. DOI: https://doi.org/10.1080/00029890.1988.11972079

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. DOI: https://doi.org/10.2140/pjm.1968.26.159

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. DOI: https://doi.org/10.1017/CBO9780511543074