My teacher told me this inequality holds but I don't know why:
Let $f$ be an holomorphic function in an open set containing the closed disk $\overline{D}(z_0,r)$. By Cauchy's Integral formula we have $$f(z_0)=\frac{1}{2\pi}\int_0^{2\pi}f\big(z_0+re^{it}\big)dt.$$ Therefore, $$|f(z_0)|\leq \frac{1}{2\pi}\int_0^{2\pi}\big|f\big(z_0+re^{it}\big)\big|dt. $$ By the way, this inequality is used for proving the Maximum Modulus Principle.