Problem 1759 (difficulty: 7/10)

Let \(\displaystyle f\) be regular on the disc \(\displaystyle |z|<1+\varepsilon\) except for finitely many poles. Assume that \(\displaystyle f(0)=1\) and that the zeros and poles of \(\displaystyle f\) inside the unit disc listed with multiplicity are \(\displaystyle \varrho_1,\varrho_2,\ldots,\varrho_n\), and \(\displaystyle p_1,p_2,\ldots,p_m\) respectively. Prove that

\(\displaystyle \frac1{2\pi}\int_{|z|=1}\log|f(z)|\cdot|dz| = \log\left|\frac{p_1p_2\ldots p_m}{ \varrho_1\varrho_2\ldots \varrho_n}\right|. \)

(If there are no zeros or poles then the respective product, that is empty, is \(\displaystyle 1\).)

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