Inequality for a holomorphic function

Question

Prove that if $f$ is a holomorphic function on the disc $D=D(z_0,R)$ and $0<r<R$ show that $$|f(z_0)|^2\leq \frac{1}{\pi r^2}||f||_D$$ Where $||f||_D=\iint_D|f(x+iy)|^2dxdy$

Answer 1

Hint: Prove that for every $0<r<R$ this formula holds :$$f(z_0)=\frac{1}{\pi r^2}\iint_{ D(z_0,R)} f(x+iy) dxdy $$. Then you take squares and using basic inequalities you get your result. This is called mean value for double integrals of a complex function.