Vol. 20, No. 3, 1967

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Inequalities for functions regular and bounded in a circle

Cecil Craig, Jr. and A. J. Macintyre

Vol. 20 (1967), No. 3, 449–454

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

wk = f(zk)(k = 1,2,3)

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 = ρe2kπi∕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).

Mathematical Subject Classification
Primary: 30.65
Received: 3 January 1966
Published: 1 March 1967
Cecil Craig, Jr.
A. J. Macintyre