Show that an entire function whose real part is positive throughout the complex plane must be a constant.
Can I get some help please?
Suppose satisfies the properties stated above, let's look at . This function is also entire since .
Since for some . Thus is bounded from below by a non zero real number.
So or in other words, is bounded from above.
Thus by Liouville's Theorem is (a non zero) constant, which implies is constant too.