Abstract
Uniform convergence of the p-Bieberbach polynomials is proved in the case of a simply connected region bounded by a piecewise quasiconformal curve with certain interior zero angles on the corner where two arcs meet.
Highlights
Finding the Riemann mapping function for a given region is a very famous and important problem for researchers
Let G be a finite region with 0 ∈ G bounded by Jordan curve L := ∂ G and let w = φ(z) be a conformal mapping of G onto the disk {w : |w| < r0} with φ(0) = 0, φ (0) = 1, where r0 is called the conformal radius of G with respect to 0
The effect of zero angles for these extremal polynomials has not yet been studied but our results show this effect
Summary
Finding the Riemann mapping function for a given region is a very famous and important problem for researchers. The best way is to approximate this function by using some extremal polynomials. 137], it is seen that there exists an extremal polynomial Pn∗ (z) furnishing to the problem (1.1). These polynomials Pn∗ (z) are determined uniquely in case p > 1 [10, p. In [14], approximation properties of p-Bieberbach polynomials were investigated in case the region was bounded by a quasiconformal curve. We will investigate the approximation rate of Bn,p(z) to the function φ in A1p-norm (Theorem 1), and by using the well-known Simonenko and Andrievskii method (see, for example, [6] and [18]), the approximation rate of Bn,p(z) to the function φ in the uniform norm will be obtained (Theorems 2–6)
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