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

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SUMMARY

The discussion centers on determining the value of the entire function f at the point (1 - 4i), given the conditions |f'(z) - (2 + 3i)| ≥ 0.00007, f(0) = 10 + 3i, and f'(7 + 9i) = 1 + i. The key conclusion is that the function cannot be constant due to the specified derivative condition. Participants are encouraged to explore the implications of these constraints on the function's behavior across the complex plane.

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  • Understanding of entire functions in complex analysis
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  • Knowledge of the concept of uniform continuity in the complex plane
  • Basic skills in solving complex equations
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Kiefer
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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)?
 
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Kiefer said:
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?
 
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.
 

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