Problem 1635 (difficulty: 7/10)

Let \(\displaystyle f\) be continuous on the closed unit disc and holomorphic inside. Let \(\displaystyle A=\max\limits_{0\le t\le\pi}|f(e^{it})|\) and \(\displaystyle {B=\max\limits_{\pi\le t\le2\pi}|f(e^{it})|}\). Show that \(\displaystyle |f(0)|\le \sqrt{AB}\).

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