Complex Variables: Prove f is Constant

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Homework Statement



Let z be a complex variable

Suppose f is an entire function and Re(f(z))\leq c for all z

Show that f is constant.
(Hint: Consider exp(f(z))

Homework Equations


possibly this: e^z=e^x(cos(y)+isin(y)) where z=x+iy

The Attempt at a Solution


I had no idea how I would show this, so I just started off trying a few things:
I first started off working with the hint to consider exp(f(z)), where exp((f(z))=ef(z)
I set g(z) equal to exp((f(z)) and because f(z) is entire, g(z) would also have to be entire
I first found a formula for the derivative of g(z) but that got me nowhere

I also tried working off the fact that Re(g(z))\leq e^ccos(Im(f(z)))
but that got me nowhere as well...

I have been thinking about this problem for so long now, and I couldn't think of a way to show that f is constant
 
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Do you know Liouville's theorem? |exp(f(z))|<=exp(c).
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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