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 $\displaystyle f(z) $ satisfies the properties stated above, let's look at $\displaystyle \frac{1}{f(z)} $. This function is also entire since $\displaystyle f(z) \neq 0 $.
Since $\displaystyle f(z) \neq 0, \; 0<a\leq |f(z)| $ for some $\displaystyle a\neq0 $. Thus $\displaystyle |f(z)| $ is bounded from below by a non zero real number.
So $\displaystyle \left|\frac{1}{f(z)}\right|\leq\frac1a $ or in other words, $\displaystyle \left|\frac{1}{f(z)}\right| $ is bounded from above.
Thus by Liouville's Theorem $\displaystyle \frac{1}{f(z)} $ is (a non zero) constant, which implies $\displaystyle f(z) $ is constant too.