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(3) f(z) C g (Z) ("f(z) is subordinate to g(z)") to mean that f(z), g(z) are both regular in I zj < 1, that g(z) is univalent there, and that every value taken by f(z) in jzj <1 is also taken by g(z) (see [2; 3]). It was shown in [1] that (4) Vn(z;f) Cf(z) (n = 1, 2, . ) for every f(z) C-K, and it was pointed out that even (5) Vl (Z;f) C V2(Z; f) C ... C f(Z) is likely, though this was not verified except for (6) fo(z) = z(1 -Z)In the following paragraphs we will show how consideration of the problem (5) leads, in a natural way, to the question of characterizing certain kinds of factor sequences (see [6]), and although we cannot decide the truth or falsity of (5), a closely related question will be completely settled (Theorem 2, infra).

Subordinating factor sequences for convex maps of the unit circle