# Thread: complex analysis entire function

1. ## complex analysis entire function

Show that an entire function whose real part is positive throughout the complex plane must be a constant.
Can I get some help please?

2. Originally Posted by dori1123
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.