Show that if f is an entire function that satisfies
|1000i + f(z)| ≥ 1000, for all z ∈ C, then f is constant.
(Hint: Consider the function g(z) = 1000/1000i+f(z) , and apply Liouville’s Theorem.)
The Attempt at a Solution
Ok, so I assume that as f is entire, then for it to be a constant, it must be bounded (Liouville’s Theorem).
Am I right in thinking that as g(z) is the reciprocal of |1000i + f(z)|, Then
1000/1000i+f(z) ≤ 1000/1000 =1
This is as far as I’ve got, I’ve sat here for hours, so any help would be very much appreciated….Thank you