Complex Analysis: What is f(1-4i)?

[Kiefer]

Suppose f is an entire function and, for every z in the complex plane, |f'(z) - (2 + 3i)| ≥ 0.00007.
Suppose also that f(0) = 10 + 3i and f'(7+ 9i) = 1 + i. What is f(1 - 4i)?

 

[SammyS]

Suppose f is an entire function and, for every z in the complex plane, |f'(z) - (2 + 3i)| ≥ 0.00007.
Suppose also that f(0) = 10 + 3i and f'(7+ 9i) = 1 + i. What is f(1 - 4i)?

What have you tried?

Where are you stuck?

 

[Kiefer]

Not really sure where to start, any help pointing me in the right direction would be great. I've done a somewhat similar problem where I proved that the function was constant, so that f equals the same value at all points, but that is not the case with this function.