Approximation by a polynomial and its derivatives on certain closed sets

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The work on the theory of approximations initiated by Weierstrass and continued by Walsh, Keldysh, and Lavrentiev, among others, has culminated in the following theorem of Mergelyan (See Mergelyan [3]): Given any compact subset C of the complex plane, which does not separate the plane, and given any continuous function f on C which is analytic interior to C, then f can be approximated uniformly on C by polynomials. This theorem leaves the following question unanswered: If fo, fl, , * * X f, are continuous functions on C, can a sequence { Pk I of polynomials be found with the property that for each integer i with 0 i 0 and 5>0 such that max {d(4[0, t], z), d(Q[t, 1], z) > A I ?(t) z I c whenever I4+(t)-z I < & Then we have