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Let $F$ be analytic on $\mathbb{H}=\{z\in\mathbb{C}:Im(z)>0\},$ continuous upto $\overline{\mathbb{H}}$ and bounded on each of the half plane $\{Im(z)\geq h>0\}.$ How to show that if $F$ satisfies $$\int_0^\infty \frac{\log(|F(x+i)|)}{1+x^2}dx=-\infty$$ then $F$ is identically zero on $\mathbb{H}?$

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  • $\begingroup$ This is Jensen's inequality in disguise: map your half-plane conformally onto the unit disk. $\endgroup$
    – Alexandre Eremenko
    Commented Nov 27, 2020 at 14:16
  • $\begingroup$ Thanks. Now I get it... $\endgroup$
    – Duplicate
    Commented Nov 27, 2020 at 16:22

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