On a "Much Underestimated" Paper of Alexander

o, M. S. Robertson, R. Nevanlinna, L. Fejand L. Bieberbach picked up on topics introduced by Alexander, and work on them has continued throughout the century. It is the purpose of this paper to analyze the contents of Alexander's work, detailing his highly intuitive arguments and how they were developed by later workers, and to follow several lines of development of this material. It is believed by some (5) that his paper was not appreciated for a long time after its publication, and the informal nature of its arguments might have led to its being described as "much underestimated" (19, p. 103). Yet the work does introduce the notions of starlike functions, close-to-convex functions, functions of bounded turning, as well as other ideas and theorems which were rediscovered as much as twenty to forty years after its appearance, in some cases apparently without knowledge of Alexander's pioneering work. We will first give some biographical background on Alexander, then summarize briefly his mathematical contributions outside of the work under discussion. We will turn next to the paper itself, analyzing its contents section by section, and include sev- eral histories of themes initiated by Alexander, and how they developed in the modern era. Our history will concentrate most heavily on the era from 1915 to the late fifties and early sixties with an occasional reference to later work.