Use of Jensen's inequality on a Riemann surface

Question

Let $f:\mathbb{C}\to \mathbb{C}$ be entire and consider the composite function $g(z):=f(\sqrt{z^2 - 1})$ on $\mathbb{C}\setminus \big ((-\infty , -1]\cup [1,\infty )\big )$ on the branch of the square root with $\operatorname{Im}\sqrt{z^2 - 1}>0$. Then $g$ can be extended across $(-\infty , -1]\cup [1,\infty )$ to the sheet with $\operatorname{Im}\sqrt{z^2 - 1}<0$.

If on that on sheet I can obtain an upper bound of $|g(z)|$ is it then legitimate to apply Jensen's formula for the number of zeros of holomorphic functions to obtain an upper bound on the number of zeros of $g$ on that sheet or is one forced to use some other method?

Answer 1

In general, Jensen's formula holds with the integral taken over both sheets (and zeros counted on both sheets). See, for example, MR1069755 Lang, S., Cherry, W. Topics in Nevanlinna theory. Lecture Notes in Mathematics, 1433. I don't know what exactly are you trying to do but if $z\to\infty$ in your problem, then it is probably more reasonable to apply Jensen's formula to $f$ rather than to $g$, using $\sqrt{z^2-1}\sim \pm z$ as $z\to\infty$.