The chain rule for $\mathcal F$-differentiation

TL;DR

This paper explores the properties of $$-differentiable functions on compact sets in the complex plane, focusing on the chain rule, its failures, and conditions for validity, with implications for algebra homomorphisms.

Contribution

It investigates the chain rule for $$-differentiability, providing conditions for its validity and extending the Faà di Bruno formula within this context.

Findings

The chain rule can fail for $$-differentiable functions.

Sufficient conditions are identified for the chain rule to hold.

The Faà di Bruno formula applies when the chain rule is valid.

Abstract

Let $X$ be a perfect, compact subset of the complex plane, and let $D^{(1)}(X)$ denote the (complex) algebra of continuously complex-differentiable functions on $X$. Then $D^{(1)}(X)$ is a normed algebra of functions but, in some cases, fails to be a Banach function algebra. Bland and the second author investigated the completion of the algebra $D^{(1)}(X)$, for certain sets $X$ and collections $\mathcal{F}$ of paths in $X$, by considering $\mathcal{F}$-differentiable functions on $X$. In this paper, we investigate composition, the chain rule, and the quotient rule for this notion of differentiability. We give an example where the chain rule fails, and give a number of sufficient conditions for the chain rule to hold. Where the chain rule holds, we observe that the Fa\'a di Bruno formula for higher derivatives is valid, and this allows us to give some results on homomorphisms between certain algebras of $\mathcal{F}$-differentiable functions.