Let f be entire and suppose that Im f (z) >= M for all z. Prove that f must be a constant function.?

Answer 1

Liouville's theorem says that an entire function that is bounded must be a constant function.

It's been a long time since I took Complex Analysis -- but I think that the quotient of two entire functions is also entire. If this is true (and you need to verify this), then

g(z) = 1/f(z)

is entire and bounded -- therefore it is constant. If g(z) is constant, then so is f(z).

If I am wrong about the quotient of entire functions -- then you need to find some other transformation of f(z) that gives you a bounded entire function.

Answer 2

This is just Liouville's boundedness theorem.