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THEOREM. Suppose that A is a complex sequence, t> 0, m is a nonnegative integer, API t(p + 1 )m for each nonnegative integer p, and f is a function such that, for each complex number z with IzI <1, f(z) EpZ=o ApzP, and f(1) is a complex number. Then the Bernstein polynomial sequence Bf has limit f on the unit disc and converges uniformly on each closed subset of the unit disc (the set of all complex numbers with modulus less than 1).

On the convergence of Bernstein polynomials for some unbounded analytic functions