Problem 1638 (difficulty: 9/10)

Let \(\displaystyle 0<r_1<r_2<r_3\) and let \(\displaystyle f\) be holomorphic on \(\displaystyle r_1<|z|<r_3\) with a continuous extension to the boundary. Prove that

\(\displaystyle \left(\max_{|z|=r_2}|f(z)|\right)^{\log(r_3/r_1)} \le \left(\max_{|z|=r_1}|f(z)|\right)^{\log(r_3/r_2)} \left(\max_{|z|=r_3}|f(z)|\right)^{\log(r_2/r_1)}. \)

(Hadamard)

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