An entire function $f(z)$ is of exponential type $\alpha$ if there exists $A$ such that $|f(z)|\leq Ae^{\alpha|z|}$ for all $z\in \mathbb C$. Given that $A=1$: how to prove that $\frac{1}{2\pi}\int_{0}^{2\pi}\log|f(re^{i\theta})|\,d\theta\leq \frac{2\alpha r}{\pi}$ for all $r>0$.
I did the following:
$\frac{1}{2\pi}\int_{0}^{2\pi}\log|f(re^{i\theta})|\,d\theta\leq \log(A)+\alpha |r|=0+r\alpha $
but I don't know how to get the $\dfrac{2}{\pi}$?