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retrostate

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## Homework Statement

Show that if f is an entire function that satisfies

|1000i + f(z)| ≥ 1000, for all z ∈ C, then f is constant.

## Homework Equations

(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