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Tchebotaröv’s extremal problem

Promarz Tamrazov (2005)

Open Mathematics

We give the complete solution of the extremal problem posed by N.G. Tchebotaröv in 20th of the last century, and we establish explicit parametric formulae for the extremals.

Uniform convergence of the generalized Bieberbach polynomials in regions with zero angles

F. G. Abdullayev (2001)

Czechoslovak Mathematical Journal

Let C be the extended complex plane; G ⊂ C a finite Jordan with 0 ∈ G ; w = ϕ ( z ) the conformal mapping of G onto the disk B 0 ; ρ 0 : = w w < ρ 0 normalized by ϕ ( 0 ) = 0 and ϕ ' ( 0 ) = 1 . Let us set ϕ p ( z ) : = ∫ 0 z ϕ ' ( ζ ) 2 / p d ζ , and let π n , p ( z ) be the generalized Bieberbach polynomial of degree n for the pair ( G , 0 ) , which minimizes the integral ∫ ∫ G ϕ p ' ( z ) - P n ' ( z ) p d σ z in the class of all polynomials of degree not exceeding ≤ n with P n ( 0 ) = 0 , P n ' ( 0 ) = 1 . In this paper we study the uniform convergence of the generalized Bieberbach polynomials π n , p ( z ) to ϕ p ( z ) on G ¯ with interior and exterior zero angles and determine its dependence on the...

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