Real total derivatives transport through the complex coordinate isometry

The inverse ${\mathrm{\Phi }}^{-1}$ is real-linear: for arbitrary $u,v\in {\mathbb{R}}^2$, choose $\zeta ,\omega \in \mathbb{C}$ with $\mathrm{\Phi }(\zeta )=u$ and $\mathrm{\Phi }(\omega )=v$, possible by [F1]. Additivity of $\mathrm{\Phi }$ gives ${\mathrm{\Phi }}^{-1}(u+v)={\mathrm{\Phi }}^{-1}(\mathrm{\Phi }(\zeta +\omega ))=\zeta +\omega ={\mathrm{\Phi }}^{-1}u+{\mathrm{\Phi }}^{-1}v$; for every $\lambda \in \mathbb{R}$, its real homogeneity gives ${\mathrm{\Phi }}^{-1}(\lambda u)={\mathrm{\Phi }}^{-1}(\mathrm{\Phi }(\lambda \zeta ))=\lambda \zeta =\lambda {\mathrm{\Phi }}^{-1}u$.

The correspondence $L\mapsto \mathrm{\Phi }\circ L\circ {\mathrm{\Phi }}^{-1}$ sends each real-linear $L$ to a real-linear map $A$ by [F1], [F2], and step 1.1. Conversely, for each real-linear $A$, the map $L={\mathrm{\Phi }}^{-1}\circ A\circ \mathrm{\Phi }$ is real-linear by the same facts, and the two constructions are inverse because ${\mathrm{\Phi }}^{-1}\circ \mathrm{\Phi }$ and $\mathrm{\Phi }\circ {\mathrm{\Phi }}^{-1}$ are identity maps. Thus this is a bijection between the real-linear maps on the two coordinate spaces.

For corresponding $L,A$ from step 2.1, put $k=\mathrm{\Phi }(h)$. Additivity gives $\mathrm{\Phi }(z+h)=\mathrm{\Phi }(z)+k$. If $r(h)=f(z+h)-f(z)-L(h)$ and $\rho (k)=f_{\mathbb{R}}(\mathrm{\Phi }(z)+k)-f_{\mathbb{R}}(\mathrm{\Phi }(z))-A(k)$, then the definition of $f_{\mathbb{R}}$ and $A(k)=\mathrm{\Phi }(L(h))$ give $\rho (k)=\mathrm{\Phi }(r(h))$. The isometry [F1] therefore gives $\mathrm{\parallel }\rho (k){\mathrm{\parallel }}_2\mathrm{/}\mathrm{\parallel }k{\mathrm{\parallel }}_2=\mathrm{\mid }r(h)\mathrm{\mid }\mathrm{/}\mathrm{\mid }h\mathrm{\mid }$ for $h\mathrm{\ne }0$; the bijective isometry also sends $h\to 0$ exactly to $k\to 0$ and sends the condition $z+h\in U$ exactly to $\mathrm{\Phi }(z)+k\in \mathrm{\Phi }[U]$. Hence the two remainder limits are equivalent.

By [F3], total differentiability of $f_{\mathbb{R}}$ at $\mathrm{\Phi }(z)$ means that some real-linear $A$ has Euclidean remainder ratio tending to zero. Steps 2.1 and 3.1 identify such maps $A$ bijectively with real-linear $L$ and identify their remainder ratios, so this condition is equivalent to existence of the stated $L$ with the complex remainder ratio tending to zero. For each corresponding pair, the Euclidean derivative is $A=\mathrm{\Phi }\circ L\circ {\mathrm{\Phi }}^{-1}$.