Polynomial Estimates over Exponential Curves in \(\mathbb C^2\)

  • Shirali KadyrovEmail author
  • Yershat Sapazhanov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)


For any complex \(\alpha \) with non-zero imaginary part we show that Bernstein-Walsh type inequality holds on the piece of the curve \(\{(e^z,e^{\alpha z}) : z \in \mathbb C\}\). Our result extends a theorem of Coman–Poletsky [6] where they considered real-valued \(\alpha \).


Bernstein-Walsh inequality Several complex variables Exponential curves 



The first author would like to thank Dan Coman for a useful discussion.


  1. 1.
    Abdullayev, F.G., Ozkartepe, N.P.: An analogue of the Bernstein-Walsh lemma in Jordan regions of the complex plane. J. Inequal. Appl. 1, 570 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brudnyi, A.: Bernstein type inequalities for restrictions of polynomials to complex submanifolds of \(\mathbb{C}^N\). J. Approx. Theory 225, 106–147 (2018)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bos, L.P., Brudnyi, A., Levenberg, N.: On Polynomial Inequalities on Exponential Curves in \(\mathbb{C}^n\). Constr. Approx. 31(1), 139–147 (2010)Google Scholar
  4. 4.
    Coman, D., Poletsky, E.A.: Measures of trancendency for entire functions. Mich. Math. J. 51(3), 575–591 (2003)CrossRefGoogle Scholar
  5. 5.
    Coman, D., Poletsky, E.: Bernstein-Walsh inequalities and the exponential curve in \(\mathbb{C}^2\). Proc. Am. Math. Soc. 131(3), 879–887 (2003)CrossRefGoogle Scholar
  6. 6.
    Coman, D., Poletsky, E.A.: Polynomial estimates, exponential curves and Diophantine approximation. Math. Res. Lett. 17(6), 1125–1136 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kadyrov, S., Lawrence, M.: Bernstein-Walsh inequalities in higher dimensions over exponential curves. Constr. Approx. 44(3), 327–338 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Neelon, T.: A Bernstein-Walsh type inequality and applications. Can. Math. Bull. 49(2), 256–264 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ransford, T.: Potential Theory in the Complex Plane. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Suleyman Demirel UniversityKaskelenKazakhstan

Personalised recommendations