Prooving that a complex function is constant

Question

Let $f(z)=u(z)+v(z)i$, $u(z) \leq 0 , \forall z \in \mathbb{C}$, and $f(z)$ is entire. Then $f(z)$ is constant. The hint is "use the Liouville theorem". I tried , but i need prove which f is limited first, and i don't know how.

Compose $f$ with an invertible transformation $T$ such that $g = T\circ f$ is bounded. — Daniel Fischer, Jun 10 '14 at 19:28

lhf · Answer 1 · 2014-06-10T19:47:11.780

3

Let $g$ be a map (a Moebius transformation) that sends the open half-plane $\Re(z)<0$ to the open unit disk biholomorphically. Then $g\circ f$ is a bounded holomorphic function and so is constant. Since $g$ is a bijection, $f$ must be constant.

edited Jun 10 '14 at 19:47

answered Jun 10 '14 at 19:31

lhf

216,483

1

Or easier, just look att $e^f$. – mrf Jun 10 '14 at 20:53
@mrf, nice! Consider adding it as an answer. – lhf Jun 10 '14 at 21:03
Nah, it's been asked and answered many times, for example here. – mrf Jun 10 '14 at 21:07

Prooving that a complex function is constant

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