I already posted this question on math.stackexchange, but got no response and was suggested to post it here.
I came across a space in an ergodic theory paper, which I am calling here a (complex) Hölder space, because of the similarity to the usual Hölder space.
Fix some open bounded convex set $U\subseteq\mathbb{C}$. For every $F:U \to B$, where $B$ is a Banach Space, set, $$\lVert F \rVert_{\infty}=\sup_{z\in U} \lVert F(z)\rVert \qquad D_{\alpha} F=\sup_{0<|z-w|<1}\left\{\frac{\lVert F(z)-F(w)\rVert}{|z-w|^\alpha} \right\}$$
If $F$ is differentiable $r$ times, define
$$\lVert F\rVert_{r,\alpha}=\sum_{k=0}^r \lVert F^{(k)}\rVert_{\infty} + D_\alpha F^{(r)}$$
In this case, the Hölder Space $\mathcal{H}_{\alpha}$ 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 found that definition here.
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. I'm looking for any kind of reference, it can be books, articles or lecture notes... That discuss some properties of this space, from a more analytical point of view, I mean, properties of the functions of these spaces, relations between the spaces for different $r$'s...
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...
I was able to find only one reference, which is the book "Holomorphic Operator Functions of One Variable and Applications" by I. Gohberg and J. Leiterer, the definition of this space is on page 33 and an image of this page can be found here. The authors call this space the Hölder-$\alpha$ continuous, in addition to the discussions in the book not being exactly what I'm looking for, the norm is slightly different, the uniform norm is taken only on $F$ and not on its derivatives.
The notation $F^{(r)}$, stands for the $r$ Frechet derivative of $F$.
