Skip to main content
SpringerLink
Log in

Approximation and Geometric Properties of Complex \(\alpha \)-Bernstein Operator

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we consider the complex form of a new generalization of the Bernstein operator, depending on a non-negative real parameter. We obtain quantitative upper estimate for simultaneous approximation, a qualitative Voronovskaja type result and the exact order of approximation. Also, we present some shape preserving properties of the complex \(\alpha \)-Bernstein operator such as univalence, starlikeness, convexity and spirallikeness.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bernstein, S.N.: Sur la convergence de certaines suites des polynomes. J. Math. Pures Appl. 15(9), 345–358 (1935)

    MATH  Google Scholar 

  2. Chen, X., Tan, J., Liu, Z., Xie, J.: Approximation of functions by a new family of generalized Bernstein operators. J. Math. Anal. Appl. 450, 244–261 (2017)

    Article  MathSciNet  Google Scholar 

  3. DeVore, R., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)

    Book  Google Scholar 

  4. Gal, S.G.: Approximation by Complex Bernstein and Convolution Type Operators, Series on Concrete and Applicable Mathematics. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2009)

    Book  Google Scholar 

  5. Graham, I., Kohr, G.: Geometric Function Theory in One and Higher Dimensions, Pure and Applied Mathematics. Marcel Dekker, New York (2003)

    MATH  Google Scholar 

  6. Kantorovich, L. V.: Sur la convergence de la suite de polynomes de S. Bernstein en dehors de l’interval fundamental. Bull. Acad. Sci. URSS 1103–1115 (1931)

  7. Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea Publ, New York (1986)

    MATH  Google Scholar 

  8. Tonne, P.C.: On the convergence of Bernstein polynomials for some unbounded analytic functions. Proc. Am. Math. Soc. 22, 1–6 (1969)

    Article  MathSciNet  Google Scholar 

  9. Wright, E.M.: The Bernstein approximation polynomials in the complex plane. J. Lond. Math. Soc. 5, 265–269 (1930)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the referees for making valuable suggestions, improving essentially the quality of the paper.

Author information

Authors and Affiliations

  1. Research Department, Turkish State Meteorological Service, 06560, Ankara, Turkey

    Nursel Çetin

Authors
  1. Nursel Çetin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nursel Çetin.

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

Çetin, N. Approximation and Geometric Properties of Complex \(\alpha \)-Bernstein Operator. Results Math 74, 40 (2019). https://doi.org/10.1007/s00025-018-0953-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-018-0953-z

Mathematics Subject Classification

Keywords

Search

Navigation