SpringerLink
Log in

On Sendov’s Conjecture

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

Sendov’s Conjecture asserts that if all the zeros of a polynomial p lie inside the closed unit disk then a closed unit disk centered at each of its zeros contains a critical point of p . In this paper, we show that the Sendov’s Conjecture holds for a polynomial if all its zeros lie on a line or a circle inside the unit disk.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Brazil)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Availability of data and material :

The manuscript has no associated data.

References

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

    Article  MATH  Google Scholar 

  2. Borcea, I.: On the Sendov conjecture for polynomials with at most six distinct roots. J. Math. Anal. Applic. 200, 182–206 (1996)

    Article  MATH  Google Scholar 

  3. Brown, J., **ang, G.: Proof of the Sendov conjecture for polynomials of degree at most eight. J. Math. Anal. Applic. 232, 272–292 (1999)

    Article  MATH  Google Scholar 

  4. Hayman, W.: Research Problems in Function Theory. Athlone, London (1967)

    MATH  Google Scholar 

  5. Miller, M.: Maximal Polynomials and the Illieff-Sendov Conjecture. Trans. Am. Math. Soc. 321(1), 285–303 (1990)

    Article  MATH  Google Scholar 

  6. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials, Oxford Science Publications; London Mathematical Society Monographs, New Series #26. Oxford University Press, (2002)

  7. Rubenstein, Z.: On a problem of Ilyeff. Pacific J. of Math. 26(1), 159–161 (1968)

    Article  Google Scholar 

  8. Schmeisser, G.: Bemerkungen zu einer Vermutung von Ilief. Math. Zeitschrift 111, 121–125 (1969)

    Article  MATH  Google Scholar 

  9. Schmeisser, G.: Zur Lage der Kritichen punkte eines polynoms. Rend. Sem. Mat. Univ. Padova 46, 405–415 (1971)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Central University of Kashmir, Ganderbal, 191201, India

    G. M. Sofi & W. M. Shah

Authors
  1. G. M. Sofi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. W. M. Shah
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. M. Sofi.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sofi, G.M., Shah, W.M. On Sendov’s Conjecture. Rend. Circ. Mat. Palermo, II. Ser 72, 493–497 (2023). https://doi.org/10.1007/s12215-021-00690-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-021-00690-y

Keywords

Mathematics Subject Classification

Search

Navigation