MAXIMAL POLYNOMIAL SUBORDINATION TO UNIVALENT-FUNCTIONS IN THE UNIT DISK

Let OMEGA subset-of C be a simply connected domain, 0 is-an-element-of OMEGA, and let P(n), n is-an-element-of N, be the set of all polynomials of degree at most n. By P(n)(OMEGA) we denote the subset of polynomials p is-an-element-of P(n) with p(0) = 0 and p(D) subset-of OMEGA, where D stands for the unit disk {z: Absolute value of z < 1}, and by OMEGA(N):= or(p is-an-element-of P(n)(OMEGA) p(D) we denote the ''maximal range'' of these polynomials. Let f be a conformal mapping from D onto OMEGA, f(0) = 0. The main theme of this note is to relate OMEGA(n) (or some important aspects of it) to the images f(s)(D), where f(s)(z):= f[(1 - s)z], 0 < s < 1. For instance we prove the existence of a universal constant co such that, for n greater-than-or-equal-to 2c0, f(c0/n)(D) subset-of OMEGA(n) subset-of OMEGA.