Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?
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7$\begingroup$ Suppose it does. Let $g(z)=exp(-f(z))$, then $g\in A(D)$. Now use the maximum principle to reach a contradiction. $\endgroup$– Onur OktayCommented Oct 2, 2022 at 9:25
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2$\begingroup$ Can't we argue even more simply that $f(0)$ is the average of $f(z)$ over $|z|=1$? $\endgroup$– Noam D. ElkiesCommented Oct 2, 2022 at 15:36
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