This paper is concerned with
functions w = f(z) regular and satisfying the inequality |f(z)| < 1 in |z| < 1. This
class of functions will be denoted E.
We determine conditions on z1,z2,z8 and w1,w2,wa for
to be possible with an f(z) of E. In particular to map the vertices of the equilateral
triangle zk= re2kπi∕3 into the vertices of another taken in the opposite direction
wk= ρe−2kπi∕3 we must have ρ ≦ r2. The extremal function associated with this
problem is w = z2. It seems convenient to discuss the fixed point if any of the
mapping of |z| < 1 into |w| < 1. We include a simple proof of the theorem of Denioy
and Wolff that if no such fixed point exists then there is a unique distinguished fixed
point on |z| = 1. We give several results restricting the position of the interior or
distinguished boundary fixed point in terms of the location of a zero of f(z) or the
value f(0).
