complex analysis entire function

**dori1123** · April 14th 2010, 02:49 PM

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

**chiph588@** · April 14th 2010, 07:25 PM

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 $f(z)$ satisfies the properties stated above, let's look at $\frac{1}{f(z)}$ . This function is also entire since $f(z) \neq 0$ .

Since $f(z) \neq 0, \; 0<a\leq |f(z)|$ for some $a\neq0$ . Thus $|f(z)|$ is bounded from below by a non zero real number.

So $\left|\frac{1}{f(z)}\right|\leq\frac1a$ or in other words, $\left|\frac{1}{f(z)}\right|$ is bounded from above.

Thus by Liouville's Theorem $\frac{1}{f(z)}$ is (a non zero) constant, which implies $f(z)$ is constant too.

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