Entire Function with Negative Imaginary Values: Proving Constantness

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An entire function f defined on the complex plane with the property that the imaginary part Imf(z) is less than or equal to zero for all z must be constant. The discussion highlights the need to utilize the Cauchy-Riemann equations and the harmonic nature of the imaginary part to demonstrate that f is bounded. The application of Liouville's Theorem is suggested as a method to conclude the proof. A hint is provided to consider the function 1/(f(z)-i) to explore its boundedness. This approach aims to clarify the relationship between the function's properties and its derivatives.
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Homework Statement


Let f:C->C be an entire function such that Imf(z) <= 0 for all z in C. Prove that f is constant.


Homework Equations


Cauchy-Riemann equations??


The Attempt at a Solution


I don't know why I haven't been able to get anywhere with this problem. I feel like I have to use the fact that Imf(z) is harmonic or satisfies the Cauchy-Riemann equations, or something like that. And then somehow show that f is bounded. From there I just apply Liouville's Theorem. But I just need a slight push in the right direction. I mean, if Imf(z) <= 0 for all z, what does that say about its derivatives? This is really frustrating.
 
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Hint: Consider the function 1/(f(z)-i). Is it bounded?
 

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