Mathematical Notes

, Volume 105, Issue 1–2, pp 216–226 | Cite as

The Möbius Transformation and Smirnov’s Inequality for Polynomials

  • E. G. GanenkovaEmail author
  • V. V. StarkovEmail author


Differential inequalities for polynomials generalizing the well-known Smirnov, Rahman, Schmeisser, and Bernstein inequalities are obtained.


Bernstein’s inequality for polynomials Smirnov’s inequality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. N. Bernstein, Extremal Properties of Polynomials and Best Approximation of Continuous Functions of a Real Variable (ONTI NKTP SSSR, Leningrad, 1937), Vol. 1 [in Russian].Google Scholar
  2. 2.
    M. Marden, Geometry of the Zeros of a Polynomial in a Complex Variable, in Math. Surveys (Amer. Math. Soc., New York, 1949), Vol. 3.zbMATHGoogle Scholar
  3. 3.
    Q. I. Rahman and G. Sshmeisser, Analytic Theory of Polynomials (Oxford Univ. Press, Oxford, 2002).Google Scholar
  4. 4.
    V. N. Dubinin, “Methods of geometric function theory in classical and modern problems for polynomials,” UspekhiMat. Nauk 67 (4 (406)), 3–88 (2012) [RussianMath. Surveys 67 (4), 599–684 (2012)].MathSciNetzbMATHGoogle Scholar
  5. 5.
    A. V. Olesov, “Differential inequalities for algebraic polynomials,” Sibirsk. Mat. Zh. 51 (4), 883–889 (2010) [Sib. Math. J. 51 (4), 706–711 (2010)].MathSciNetzbMATHGoogle Scholar
  6. 6.
    V. N. Dubinin, “Polynomials with critical values on intervals,” Mat. Zametki 78 (6), 827–832 (2005) [Math. Notes 78 (6), 768–772 (2005)].MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. N. Bernstein, “Sur la limitation des dérivées des polynomes,” C. R. Acad. Sci. Paris 190, 338–341 (1930).zbMATHGoogle Scholar
  8. 8.
    V. I. Smirnov and N. A. Lebedev, Constructive Theory of Functions of a Complex Variable (Nauka, Moscow, 1964) [in Russian].zbMATHGoogle Scholar
  9. 9.
    Q. I. Rahman, “Functions of exponential type,” Trans. Amer. Math. Soc. 135, 295–309 (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. L. Wali, W. M. Shah, and A. Liman, “Inequalities concerning B–operators,” Probl. Anal. Issues Anal. 5 (23) (1), 55–72 (2016).Google Scholar
  11. 11.
    I. Qasim, A. Liman, and W. M. Shah, “Refinement of some inequalities concerning to Bn–operator of polynomials with restricted zeros,” Period. Math. Hung. 74 (1), 1–10 (2017).CrossRefzbMATHGoogle Scholar
  12. 12.
    A. Liman, R. N. Mohapatra, and W. M. Shah, “Inequalities for polynomials does not vanishing in a disk,” Appl. Math. Comput. 218 (3), 949–955 (2011).MathSciNetzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Petrozavodsk State UniversityPetrozavodskRussia

Personalised recommendations