Can someone explain the following Cauchy integral estimate?
\begin{eqnarray*} \left| \frac{\partial f}{\partial z_r} (z) \right| &=& \left| \frac{1}{(2\pi i)^n} \int_{\left| w_{\nu_j} - \zeta_j \right| = r_{\nu} + \delta_{\nu}} \frac{f(\zeta)}{(\zeta_1 - z_1) \cdots (\zeta_r - z_r)^2 \cdots (\zeta_n - z_n)} d\zeta_1 \wedge \cdots \wedge d\zeta_n \right| \\ & \leq & \frac{M}{\delta_{\nu}} \left( 1 + \frac{r_{\nu}}{\delta_{\nu}} \right)^n, \end{eqnarray*} where $\left| f \right| \leq M$.
It's not too bad.
Simply consider that \begin{eqnarray*} \left| \frac{\partial f}{\partial z_r}(z) \right| &=& \left| \frac{1}{(2\pi i)^n} \int_{\left| w_{\nu_j} - \zeta_j \right| = r_{\nu} + \delta_{\nu}} \frac{f(\zeta)}{(\zeta_1 - z_1) \cdots (\zeta_r - z_r)^2 \cdots (\zeta_n - z_n)} d\zeta_1 \cdots d\zeta_n \right| \\ & \leq & \frac{1}{(2\pi)^n} (2\pi)^n \frac{(r_{\nu} + \delta_{\nu})^{n}}{\delta_{\nu}^{n+1}} \max_{\left| w_{\nu_j} - \zeta_j \right| = r_{\nu} + \delta_{\nu}} \left| f(\zeta) \right| \\ & \leq & \frac{M}{\delta_{\nu}} \left( 1 + \frac{r_{\nu}}{\delta_{\nu}} \right)^n. \end{eqnarray*} Note that the first inequality uses the fact that \begin{eqnarray*} \left| \zeta_{\nu} - z_{\nu} \right| &=& \left| \zeta_{\nu} - w_{\nu} + w_{\nu} - z_{\nu} \right| \\ & \geq & \left| \zeta_{\nu} - w_{\nu} \right| - \left| z_{\nu} - w_{\nu} \right| \\ & = & r_{\nu} + \delta_{\nu} - \left| z_{\nu} - w_{\nu} \right| \\ & \geq & r_{\nu} + \delta_{\nu} - r_{\nu} = \delta_{\nu}. \end{eqnarray*}