Approximation by Polynomials Subordinate to a Univalent Function Article Swipe
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· 1970
· Open Access
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· DOI: https://doi.org/10.2307/1995047
· OA: W4245622767
This paper is concerned with approximating a function /(z) analytic and univalent in the unit disk £={z: \z\ < 1} by polynomials which are also univalent in £.We are interested in such approximations where exactly one polynomial of each degree n, n â 1, is used and such that the polynomials are monotonically subordinate to each other.Recall that/(z) is called subordinate to g(z) in £ if both functions are analytic in £ and if there exists a function <p(z) analytic in £ and satisfying |cp(z)|<l, <p(0) = 0 such that /(z)=g(<p(z)).The existence of such a function ç>(z) is implied by the conditions that giz) is univalent in £,/(0) = g(0) and/(£)^g(£).If/(z) is subordinate to g(z) in £ we will write/(z)^g(z) in £.Theorem 1 asserts that if /(z) is analytic and univalent in £ then there exists a sequence {p"iz)}, «=1,2,..., such that />n(z) is a polynomial of degree n which is univalent in E, Pi(z) c Pi(z) <= p3(z) <=■■• and pniz) -¡*f(z) as n -> co.This convergence is uniform in each compact subset of £.The idea of proving this is to appropriately relate /(z) to the partial sums 5n(z) of the power series for/(z).Although s"iz) are not, in general, univalent in £ (they are in \z\ <£ [12]) they are univalent in \z\ <r, 0<r< 1, for all large n once r is given.This eventually leads to a chain of univalent polynomials Pnx(z) <= Pn2(z) c pnai¿) CZ... such that Pnfz) has degree n¡, nx < n2 < n3 < ■ ■ ■, and pn/iz) ->/(z) as / -*■ oo.These polynomials are so related that it is still possible to find polynomials of the remaining degrees which fill in the chain and such that pjf) -^/(z).This method is equally adaptable when /(z) maps £ one-to-one onto a convex domain in order to produce a similar chain of convex, univalent polynomials.In fact, our argument may be applied to any one of a number of classes of univalent functions.
