Suppose f C rightarrow C is analytic on C and that fz 1 f

Suppose f: C rightarrow C is analytic on C and that | f(z) | > 1 for all z in C. Show that f must be constant on C. (Suggestion: What can you say about 1/f?)

Solution

SOLUTION:

Since |f(z)|>1, f(z)0 for all z in C.

Therefore, 1/f(z) is analytic in C and hence an entire function.

Also, |f(z)|>1 implies,1/|f(z)| <1for all z in C. That is, 1/f is bounded over C.

Thus 1/f is a entire bounded function. So by Liouville\'s Theorem, 1/f is constant andand hence f is constant.