Regions of Variability for Univalent Functions

Let S be the standard class of univalent functions in the unit disk, and let So be the class of nonvanishing univalent functions g with g(0) = 1. It is shown that the regions of variability {g(r): g £ So} and {(1 — r2)f'(r) : f £ S} are very closely related but are not quite identical. Let 5 be the class of functions / analytic and univalent in the unit disk D, with /(0) = 0 and /'(O) = 1. Let So be the class of analytic univalent functions g for which g(z) ^ 0 in D and ^(0) — 1. In this paper we compare the regions of variability W(t) = {/'(?):/ €5} and W0(ç) = {g(ç): g G S0} at a fixed point ç G D. Both regions depend only on |ç|, so it suffices to consider W[r) and W0{r) for 0 < r < 1. Initially there is no reason to expect the regions W(r) and VFo(r) to be related at all. There is only the superficial connection suggested by the nonvanishing of the derivative of a univalent function. However, we shall offer persuasive evidence in support of the conjecture that W0{r) = {l-r2)W(r), 0 < r < 1. For convenience, we let W(r) = (1 — r2)W(r). In §1 we motivate the conjecture that Wo(r) = W(r), and we present some analytical evidence in its favor. In §2 we show how the boundaries of the two regions can be calculated, and we display samples of numerical data which seem to leave little doubt that the conjecture is true. In §3 we apply a variational method to prove that Wo(r) C W(r). Then comes the surprise: we show in §4 that the conjecture is false. Thus although the two regions Wo(r) and W{r) share various features and appear "numerically identical", one is in fact a proper subset of the other. 1. Analytical evidence. The sharp estimates for |/'(r)| are given by the classical distortion theorem with equality for the Koebe function k(z) = z(l — z)~2 and its rotation — fc(—z). The corresponding inequality for Sq is [13, 8] _ e^î^^(\^)2' if*. Received by the editors April 5, 1985. Presented to the American Mathematical Society in San Diego, California, on November 10, 1984. 1980 Mathematics Subject Classification. Primary 30C55; Secondary 30C70.