Complex Hölder Space Usually the Hölder Space is defined for functions of $\mathbb{R}^n$ to $\mathbb{R}$ which is easily extended to any Banach space $B$. But I came across the definition with the domain being $\mathbb{C}$ i.e.  let $U\subseteq\mathbb{C}$ define,
$$\lVert F \rVert_{\infty}=\sup_{z\in U} \lVert F(z)\rVert \qquad D_{\alpha} F=\sup_{0<|z-w|<1}\left\{\frac{\lVert F(z)-F(w)\rVert}{|z-w|^\alpha} \right\}$$
If $F$ is differentiable $r$ times, set
$$\lVert F\rVert_{r,\alpha}=\sum_{k=0}^r \lVert F^{(k)}\rVert_{\infty} + D_\alpha F^{(r)}$$
In this case, the Hölder Space is the space of all $F:U\to B$ which are differetiable $r$-times in $U$ and for which $\lVert F\rVert_{r,\alpha}$ is finite.
I'm having trouble finding a book that deals with this space. I can see that the real and complex case share several properties, but I would like some reference. It can also be a reference for the case where the image is any Banach space, as I also have a few reference in this situation.
I imagine this space can be thought of when considering $\mathbb{R}^2$, but I would like a reference to see certain specific features when we are dealing with complex functions....
The notation $F^{(r)}$, stands for the $r$ frechet derivative of $F$.

I think Zhu talks about it in "Spaces of Holomorphic Functions in the Unit Ball". It is refered as Lipschitz Spaces.
