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Bound between mean value and extremal value of holomorphic function

Source: Alibaba Global Math Competition 2021, Problem 8

July 4, 2021
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Problem Statement

Let f(z)f(z) be a holomorphic function in {∣z∣≤R}\{\vert z\vert \le R\} (0<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
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