ALIBABA 2021 Problem 8

Let $f(z)$ be a holomorphic function in $\{\vert z\vert \le R\}$ ( $0<R<\infty$ ). Define $$M(r,f)=\max_{\vert z\vert=r} \vert f(z)\vert, A(r,f)=\max_{\vert z\vert=r} \text{Re}\{f(z)\}.$$ Show that $$M(r,f) \le \frac{2r}{R-r}A(R,f)+\frac{R+r}{R-r} \vert f(0)\vert, \forall 0 \le r<R.$$