Boundedness of derivatives and anti-derivatives of holomorphic functions as a rare phenomenon

TL;DR

This paper demonstrates that, on simply connected domains, most derivatives and anti-derivatives of holomorphic functions tend to be unbounded, revealing a rare boundedness phenomenon and exploring universality in Taylor partial sums.

Contribution

It establishes that bounded derivatives and anti-derivatives are rare for holomorphic functions on simply connected domains and discusses universality properties of Taylor partial sums.

Findings

Derivatives and anti-derivatives of generic holomorphic functions are unbounded.

A similar unboundedness result applies to Taylor partial sums as functions of the center.

Universality of Taylor partial sums is discussed.

Abstract

In this article we prove a general result which in particular suggests that, on a simply connected domain in C, all the derivatives and anti-derivatives of the generic holomorphic function are unbounded. A similar result holds for the operator of partial sums of the Taylor expansion with center z at 0, seen as functions of the center z. We also discuss a universality result of these operators.