Let $f(z)$ be a non-constant analytic function in an open region $\Omega \subset \mathbb{C}$. For every $z \in \Omega$ with a ball $B_r(z) \subset \Omega$, there exists a $z^{\prime} \in B_r(z)$ such that $\left|f\left(z^{\prime}\right)\right|>|f(z)|.$ ###### Proof By Cauchy's integral theorem, for every $r>0$ such that $B_{r}(z) \subset \Omega$ we have that $\begin{align} f(z) &= \frac{1}{2\pi i}\oint_{\partial B_{r}(z)} \frac{f(w)}{w-z} \, dw =\frac{1}{2\pi i } \int_{0}^{2\pi} \frac{f(z+re^{i\theta})}{re^{i\theta}} i re^{i\theta} \, d\theta = \frac{1}{2\pi } \int_{0}^{2\pi} f(z+re^{i\theta}) \, d\theta. \end{align}$ This is known as the mean value theorem. Noe for all