Essential Angular Derivatives and Maximum Growth of Koenigs Eigenfunctions

We introduce the notion ofessential angular derivativefor functionsφmapping the open unit diskUholomorphically into itself. After exploring some of its basic properties, we show how the essential angular derivative ofφdetermines the maximum growth rate of the Koenigs eigenfunctionσforφwhenφhas an attractive fixed point inU. Our work answers some questions about growth of Koenigs functions recently posed by Pietro Poggi-Corradini.