On the definition of a close-to-convex function

Abstract

The standard definition of a close-to-convex function involves a complex numerical factor eiβ which is on occasion erroneously replaced by 1. While it is known to experts in the field that this replacement cannot be made without essentially changing the class, explicit reasons for this fact seem to be lacking in the literature. Our purpose is to fill this gap, and in so doing we are lead to a new coefficient problem which is solved for n=2, but is open for n>2.