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.

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}}-\overline{z_{2}} \end{array}\right), \end{equation} where $z(\tau) = z_2 + \tau (z_1 - z_2)$, $\overline{z_1} - \overline{z_2}$ denotes the conjugate of $z_1 -z_2$ and $\nabla f(z(\tau))$ is the Wirtinger Jacobian.