Suppose that $f ( z )$ is analytic, and satisfies the condition $\left| f ( z ) ^ { 2 } - 1 \right| = | f ( z ) - 1 | \cdot | f ( z ) + 1 | < 1$ on a non-empty connected open set $U$. Then,\\
(a) $f$ is constant.\\
(b) The imaginary part of $f , \operatorname { Im } ( f )$, is positive on $U$.\\
(c) The real part of $f , \operatorname { Re } ( f )$, is non-zero on $U$.\\
(d) $\operatorname { Re } ( f )$ is of fixed sign on $U$.