The statement of the theorem is as follows:
Theorem(10.3.4): Suppose $f$ is a holomorphic function on the upper half-plane satisfying:
- $f(\tau+2)=f(\tau)$,
- $f(-1/\tau)=f(\tau)$,
- $f(\tau)$ is bounded. Then $f$ is a constant.
Assume in contrary $f$ is not constant, based on the periodicity and relation (2), we can restrict our attention to the fundamental domain $\mathcal{F}$ of the shape
.
Using properties 1-3, one will be able to show that by maximum modulus principle, $$\lim_{\text{Im}(\tau)\to\infty}|f(\tau)|<\sup_{\tau\in\mathcal{F}}|f(\tau)|,\;\;\;\lim_{\text{Im}(\tau)\to\infty}|f(1-1/\tau)|<\sup_{\tau\in\mathcal{F}}|f(\tau)|.$$ Then, in the end of the proof, they seems claimed the second inequality above will implies that $f$ will attains maximum in the interior of the upper half-plane, contradicting the maximum modulus principle.
I am confused at this point. Why was that?