Let $f$ be holomorphic on an environment of $D_{R}(a)$. Prove $|f(a)|\leq \frac{1}{2\pi}\int_{0}^{2\pi}{|f(a+Re^{it})|dt}$

Question

I don't know how to proceed with this one:

Let $f$ be holomorphic on an environment of $D_{R}(a)$. Prove $|f(a)|\leq \frac{1}{2\pi}\int_{0}^{2\pi}{|f(a+Re^{it})|dt}$

Could anyone give me a hint? It has another part, but I need to solve this first to solve the second statement.

Thanks for your time.

Answer 1

Use the fact that$$f(a)=\frac1{2\pi i}\int_\gamma\frac{f(z)}{z-a}\,\mathrm dz,$$with $\gamma(t)=a+Re^{it}$ ($t\in[0,2\pi]$).