Approximation Theory and Its Applications

, Volume 17, Issue 1, pp 97–105 | Cite as

On the Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with K-Quasiconformal Boundary

  • Abdullah Cavus
  • Fahreddin G. Abdullayev


Let G be a finite domain in the complex plane with K-quasicon formal boundary, z0be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r0>0 and centered at the origin 0, normalized by ϕ(z0) = 0 and ϕ(z0) = 1. Let us set\(\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta \), and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G,z0) that minimizes the integral\(\iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x }\)in the class\(\mathop \prod \limits_n \)of all polynomials of degreenand satisfying the conditions P n (z0) = 0 and P n (z0) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to ϕ p (z) on\(\bar G\)in case of\(p > 2 - \frac{{K^2 + 1}}{{2K^4 }}\).


Complex Plane Conformal Mapping Uniform Convergence Formal Boundary Finite Domain 
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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Abdullah Cavus
    • 1
  • Fahreddin G. Abdullayev
    • 2
  1. 1.Faculty of Arts & Sciences, Department of MathematicsKaradeniz Technical UniversityTrabzonTurkey
  2. 2.Institute of Mathematics & MechanicsAcademy of Sciences of Azerbaijan RepublicBakuAzerbaijan

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