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This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function theory and potential theory.

We extend the results on the uniform convergence of Bieberbach polynomials to domains with certain interior zero angles (outward pointing cusps), and show that they play a special role in the problem. Namely, we construct a Keldysh-type example on the divergence of Bieberbach polynomials at an outward pointing cusp and discuss the critical order of tangency… (More)

Let Ω be a domain in the complex plane C whose complement E = C \ Ω, where C = C ∪ {∞} is a subset of the real line (i.e. Ω is a Denjoy domain). If each point of E is regular for the Dirichlet problem in Ω, we provide a geometric description of the structure of E near infinity such that the Martin boundary of Ω has one or two " infinite " points.

We study the uniform approximation of the canonical conformal mapping, for a Jordan domain onto the unit disk, by polynomials generated from the partial sums of the Szeg˝ o kernel expansion. These polynomials converge to the conformal mapping uniformly on the closure of any Smirnov domain. We prove estimates for the rate of such convergence on domains with… (More)

We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, k ≥ 1) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E. Furthermore, the error in this approximation is decaying as e −cn α strictly inside E , where c and α are positive constants… (More)