## Uniform Approximations to Functions of a Complex Variable |

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absolute constant analogous analytic functions arbitrary continuum argz asymptotic best approximations boundary points bounded closed set Carleman Cauchy Cauchy kernel circle of radius closed region condition of order conformal mapping connected complement consider contains the point continuous functions converges uniformly curve denote dense sets depend derivative diameter divide the plane equation estimate exceed exists an integral exponential type fg(z finite number follows function f(z harmonic measure hence holomorphic integral function G(z interior points interval Jordan arcs Lemma Lipschitz condition M. V. Keldys modulus of continuity numbers pn(f obtained point at infinity poles positive number problem proof properties proposition proved rapidity of decrease rational function R(z real axis region bounded regions complementary S. N. Bernstein satisfies the conditions satisfying the inequality sector sequence series of polynomials situated strip Imz Theorem 1.3 theory of best tion uniform approximation uniformly convergent up(z whole plane