Bounded holomorphic functions with given maximum modulus on all circles

Abstract

We study Q C Cd, a circular, bounded, strictly convex domain with c2 boundary. Let g and h be continuous functions on 9Q with Jg(z)j < h(z) = h(Az) for z E 8Q and JAI = 1. First we prove that h can be ap proximated by the maximum modulus values of K homogeneous polynomials, where K is independent from h. Next we construct fl E A(Q) such that max I (g + fl) (Az) I = h(z) for z E (9Q. Moreover we can choose f2 e G(Q) with Jf2*(z)I = h(z) for almost all z E &Q and max1X1<l 1f2(Az)I = h(z) for all z E 9Q.