Skip to main content
SpringerLink
Log in
Book cover

Approximation and Computation pp 61–85Cite as

Extremal Problems for Polynomials in the Complex Plane

  • Chapter
  • First Online:

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 42))

Abstract

This is a survey on some particular polynomial problems that are related to complex analogs of Rolle’s theorem or to the Bernstein majorization theorem that implies the well-known estimate for the derivative of a complex polynomial on the disk. The main topic, however, is Sendov’s conjecture about the critical points of algebraic polynomials. Despite the numerous attempts to verify the conjecture, it is not settled yet and remains as one of the most challenging problems in the analytic theory of polynomials. We also discuss the mean value conjecture of Smale and point out to certain relation between these two famous open problems. Finally, we formulate a conjecture that seems to be a natural complex analog of Rolle’s theorem and contains as a particular case Smale’s conjecture.

Deceased April 8, 2009

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bernstein, S.N.: Sur une propriété de polinômes. Comm. Soc. Math. Kharkow Sér. 2. 14, 1-2, 1–6 (1913) In: Collected Works, Volume 1, Izd. AN SSSR, Moscow, 146–150 (1952)

    Google Scholar 

  2. Bernstein, S.N.: Sur la limitation des dérivées des polynomes. C. R. Acad. Sci. Paris 190, 338–341 (1930)

    MATH  Google Scholar 

  3. Bojanov, B.: The conjecture of Sendov about the critical points of polynomials. Fiz.-Mat. Spisanie, 140–150 (1984)

    Google Scholar 

  4. Bojanov, B.: Markov-type inequalities for polynomials and splines. 31–90. In: Approximation Theory X: Abstract and Classical Analysis, Charles Chui, L.L. Schumaker, and J. Stöckler (eds.), Vanderbilt University Press, Nashville, TN (2002)

    Google Scholar 

  5. Bojanov, B., Rahman, Q.I., Szynal, J.: On a conjecture of Sendov about the critical points of a polynomial. Math. Z. 190, No 2, 281–186 (1985)

    Google Scholar 

  6. Bojanov, B., Naidenov, N.: Majorization of polynomials on the plane. II. East J. Math. 12, No 2, 189–202 (2006)

    Google Scholar 

  7. Borcea, J.: Maximal and inextensible polynomials. Preprint: arXiv math.CV/0601600vl, May 29, 2006.

    Google Scholar 

  8. de Bruijn, N.G.: Inequalities concerning polynomials in the complex domain. Nederl. Akad. Wetensch. Indag. Math. 9, 591–598 (1947)

    Google Scholar 

  9. de Bruijn, N.G., Springer, T.A.: On the zeros of a polynomial and of its derivative II. Nederl. Akad. Wetensch. Indag. Math. 9, 264–270 (1947)

    Google Scholar 

  10. Conte, A., Fujikawa, B., Lakic, N.: Smale’s mean value conjecture and the coefficients of univalent functions. Proc. Am. Math. Soc. 135, No 10, 3295–3300 (2007)

    Google Scholar 

  11. Dimitrov, D.K.: A refinement of the Gauss-Lucas theorem. Proc. Am. Math. Soc. 126, 2065–2070 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dimitrov, D.K.: Smale’s conjecture on mean values of polynomials and electrostatics. Serdica Math. J. 33, No 4, 399–410 (2007)

    Google Scholar 

  13. Dryanov, D.P., Rahman, Q.I.: On a polynomial inequality of E. J. Remez. Proc. Am. Math. Soc. 128, 1063–1070 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Duffin, R.J., Schaeffer, A.C.: A refinement of an inequality of the brothers Markoff. Trans. Am. Math. Soc. 50, 517–528 (1941)

    MATH  MathSciNet  Google Scholar 

  15. Erdös, P.: Some remarks on polynomials. Bull. Am. Math. Soc. 53, 1169–1176 (1947)

    Article  MATH  Google Scholar 

  16. Goodman, A.W., Rahman, Q.I., Ratti, J.S.: On the zeros of a polynomial and its derivative. Proc. Am. Math. Soc. 21, 273–274 (1969)

    MATH  MathSciNet  Google Scholar 

  17. Malamud, S.M.: Inverse spectral problem for normal matrices and generalization of the Gauss-Lucas theorem. arXiv:math.CV/0304158v3 6Jul2003.

    Google Scholar 

  18. Marden, M.: Geometry of Polynomials. Am. Math. Soc. Providence (1966)

    Google Scholar 

  19. Marden, M.: Much ado about nothing. Am. Math. Mon. 83, 788–789 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  20. Marden, M.: Conjectures on the critical points of a polynomial. Am. Math. Mon. 90, 267–276 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  21. Markov, V.A.: On the functions of least deviation from zero in a given interval. St. Petersburg (1892) (in Russian); German translation with shortenings: W. Markoff: Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen. Math. Ann. 77, 213–258 (1916)

    Google Scholar 

  22. Meir, A., Sharma, A.: On Ilief’s conjecture. Pacific J. Math. 31, 459–467 (1969)

    MATH  MathSciNet  Google Scholar 

  23. Milovanović, G.V., Mitrinović, D.S., Rassias, Th.M.: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  24. Nikolov, G.: Inequalities of Duffin-Schaeffer type. SIAM J. Math. Anal. 33, 686–698 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Nikolov, G.: An extension of an inequality of Duffin and Schaeffer. Constr. Approx. 21, 181–191 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  26. Obrechkoff, N.: Zeros of Polynomials. Bulgarian Academic Monographs (7), Marin Drinov Academic Publishing House, Sofia (2003) (English translation of the original text, published in Bulgarian in 1963).

    Google Scholar 

  27. Pereira, R.: Differentiators and the geometry of polynomials. J. Math. Anal. Appl. 285, 336–348 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Oxford Science Publications, Clarendon, Oxford (2002)

    MATH  Google Scholar 

  29. Rivlin, T.J.: Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory. Second edition, Wiley, New York (1990)

    MATH  Google Scholar 

  30. Schmeisser, G.: On Ilieff’s conjecture. Math. Z. 156, 165–175 (1977),

    Article  MATH  MathSciNet  Google Scholar 

  31. Schmeisser, G.: Bemerkungen zu einer Vermutung von Ilieff. Math. Z. 111, 121–125 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  32. Schmeisser, G.: The conjectures of Sendov and Smale. pp. 353–369. In: Approximation Theory: A volume dedicated to Blagovest Sendov (B. Bojanov, Ed.), DARBA, Sofia (2002)

    Google Scholar 

  33. Schmieder, G.: A proof of Sendov’s conjecture. Preprint: arXiv:math.CV/0206173 v6 27 May 2003

    Google Scholar 

  34. Schoenberg, I.J.: A conjectured analogue of Rolle’s theorem for polynomials with real and complex coefficients. Am. Math. Mon. 93, 8–13 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  35. Sendov, Bl.: Hausdorff geometry of polynomials. East J. Approx. 7, No 2, 123–178 (2001)

    Google Scholar 

  36. Sendov, Bl.: Complex analogues of the Rolle’s theorem. Serdica Math. J. 33, No 4, 387–398 (2007)

    Google Scholar 

  37. Sendov, Bl., Marinov, P.: Verification of Smale’s mean value conjecture for n ≤ 10. C. R. Acad. Bulg. Sci. 60, No 11, 1151–1156 (2007)

    Google Scholar 

  38. Smale, S.: The fundamental theorem of algebra and complexity theory. Bull. Am. Math. Soc. (New Series) 4, 1–36 (1981)

    Google Scholar 

  39. Schaeffer, A.C., Szegö, G.: Inequalities for harmonic polynomials in two and three dimensions. Trans. Am. Math. Soc. 50, 187–225 (1941)

    MATH  Google Scholar 

  40. Szegö, G.: Components of the set \(\vert (z - {z}_{1})\cdots (z - {z}_{n})\vert \leq 1\). Am. Math. Mon. 58, No 9, p. 639 (1951)

    Google Scholar 

  41. Tischler, D.: Critical points and values of polynomials. J. Complex. 5, 438–456 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  42. Walsh, J.L.: Lemniscates and equipotential curves of Green’s function. Am. Math. Mon. 42, 1–17 (1935)

    Article  Google Scholar 

  43. Walsh, J.L.: On the convexity of the ovals of lemniscates. pp. 419-423. In: Studies in Mathematical Analysis and Related Topics, Stanford University Press, Stanford, Calif. (1962)

    Google Scholar 

Download references

Acknowledgements

The author is grateful to his colleagues Lozko Milev and Nikola Naidenov for their help in performing computer calculations confirming Conjecture 1 for polynomials of small degree.

This work was supported by the Sofia University Research Grant # 135/2008 and by Swiss-NSF Scopes Project IB7320-111079.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Sofia, Blvd. James Boucher 5, 1164, Sofia, BULGARIA

    Borislav Bojanov

Authors
  1. Borislav Bojanov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Dept. Computer Science, Purdue University, N. University Street 305, West Lafayette, 47907-2107, Indiana, USA

    Walter Gautschi

  2. Dipto. Matematica, Università Basilicata, Contrada Macchia Romana, Potenza, 85100, Italy

    Giuseppe Mastroianni

  3. Zagoras Street 4, Athens, 151 25, Greece

    Themistocles M. Rassias

Additional information

Dedicated to Gradimir V. Milovanović on the occasion of his 60th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Bojanov, B. (2010). Extremal Problems for Polynomials in the Complex Plane. In: Gautschi, W., Mastroianni, G., Rassias, T. (eds) Approximation and Computation. Springer Optimization and Its Applications, vol 42. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-6594-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation