Vol. 20, No. 3, 1967

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 323: 1
Vol. 322: 1  2
Vol. 321: 1  2
Vol. 320: 1  2
Vol. 319: 1  2
Vol. 318: 1  2
Vol. 317: 1  2
Vol. 316: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
Inequalities for functions regular and bounded in a circle

Cecil Craig, Jr. and A. J. Macintyre

Vol. 20 (1967), No. 3, 449–454
Abstract

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
Milestones
Received: 3 January 1966
Published: 1 March 1967
Authors
Cecil Craig, Jr.
A. J. Macintyre