# Imaginary part of ff is positive on U

by 1256   Last Updated April 16, 2018 07:20 AM

Suppose that $f(z)$ is analytic and satisfies the condition $\vert f(z)^2 - 1\vert=\vert f(z)-1 \vert \vert f(z)+1\vert<1$ on a nonempty connected open set $U$ then conclude that the below statement is false

1) $f$ is constant.

2) Imaginary part of $f$ is positive on $U$

Hint. Take $f(z)=z$ and $U=\{z\in\mathbb{C}: |z-1|<r\}$ with $r>0$ sufficiently small.