An approximation theorem for biholomorphic functions on $D\sp{n}$

Abstract

Let F be a biholomorphic mapping of the polydisk Dn into Cn. We construct a sequence of polynomial mappings 1P A such that each P' is subordinate to P + , each Pi is subordinate to F and the P i converge uniformly on compacta to F. The polynomials P are biholomorphic. Introduction. Le t DT be the disk in the complex plane C with center at the origin and radius r > 0 (D1 = D). MacGregor Li] has shown that if f is a schlicht mapping of D into C then there exists a sequence of schlicht polynomials IP to.o *1 j=11 each P has degree j, such that Pj converges uniformly to f on compacta and such that P, is subordinate to Pj+ 1 for each j = 1, 2,. A second result is that if f is a convex schlicht mapping of D into C, then the IPV in the above result can be chosen to be convex schlicht polynomials. A close scrutiny of the proofs of these results show that they depend principally upon the facts that C is a normed linear space and the fact that f is a homeomorphism. We extend these results to the following case. Let Dn be the n fold product of D and assume F is a biholomorphic mapping of Dn into Cn, F(O) = 0. Then there exists a sequence of polynomial mappings IP .1, which are biholomorphic, and which converge uniformly to F on compacta. Further, each P. is subordinate to P. for j=1, 2, 3,.. Using a result of T. J. Suffridge we can also show that if F(Dn) is convex in Cn then each P. can be chosen so that P i(Dn) is convex. Notation and definition. Let D. denote the disk in the complex plane with center at the origin and radius r > 0, Dn is the n-fold product of such disks and D is the closure of such a disk. If r= 1 we omit the subscript. A point Z in 7 cn will be written as Z = (z . z ) z. CC, and a mapping F from Dn into n Cn as F(Z)= (f1(Z),.--, fn(Z))= W. If f1(Z) is holomorphic on Dn, then f.(Z)= m ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I k=O h i>(Z) which hl k are homogeneous polynomials of degree k. For N a positive integer we let fb,N(Z) = _ = hj,(Z) and FN(Z) = (f1,N(Z), f2,N(Z), ... fn,N(Z)). Whenever a sequence of mappings FN converge uniformly on compacta to a mapping F we will write FN(Z) =* F(Z). A mapping F (from Dn into Cn) Received by the editors January 18, 1972. AMS (MOS) subject classifications (1970). Primary 32A05, 30A36.