The mean value theorem for vector-valued function in the real domain $f: \mathcal{R}^n \rightarrow \mathcal{R}^d$ can be expressed as
\begin{equation}
f(x)-f(y)=\int_{0}^{1}\nabla f(x(\tau))d \tau \cdot (x -y),
\end{equation}
where $x(\tau) = y + \tau (x - y)$. 

I wonder if there exists similar results for vector-valued functions in the complex domain. Suppose $f:\mathcal{C}^n \rightarrow \mathcal{C}^d$, do we have
\begin{equation}
f(z_1)-f(z_2)=\int_{0}^{1}\nabla f(z(\tau))d \tau \cdot \left[\begin{array}{c}
z_{1}-z_{2}\\
\overline{z_{1}-z_{2}}
\end{array}\right],
\end{equation}
where $z(\tau) = z_2 + \tau (z_1 - z_2)$, $\overline{z_1 - z_2}$ denotes the conjugate of $z_1 -z_2$ and $\nabla f(z(\tau))$ is the Wirtinger Jacobian.