What might I use to show that an entire function with positive real parts is constant?

complex-analysis

So the question asks me to prove that an entire function with positive real parts is constant, and I was thinking that this might somehow be related to showing an entire bounded function is constant (Liouville's theorem), but are there any other theorems that might help me prove this fact?


The other three answers are overkill to me.. Simply consider $e^{-f}$ if $f$ is your function. Is it bounded?


It isn't a nonconstant polynomial, by the fundamental theorem of algebra. It doesn't have an essential singularity at infinity, by the Casorati-Weierstrass theorem. What other possibilities are there?

Alternatively, if you add $1$, you get a function satisfying $|g(z)|\geq 1$ for all $z$. What can you say about the reciprocal of $g$?

Related

Determinant of rank-one perturbation of a diagonal matrix
Complex integral over a circle
Show that $\int^1_0 \frac{\tanh^{-1} x}{x} \ln [(1 + x)^3 (1 - x)] \, dx = 0$
Uniform continuity and boundedness
How to prove $\sum\limits_{n=1}^\infty\frac{\sin(n)}n=\frac{\pi-1}2$ using only real numbers.
For which $n$, $G$ is abelian?
Categorial definition of subsets
How prove this $(p-1)!\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{p-1}\right)\equiv 0\pmod{p^2}$
Number of integer triplets $(a,b,c)$ such that $a<b<c$ and $a+b+c=n$
Looking for a Calculus Textbook
Do Boolean rings always have a unit element?
$\mathbb{Z}[\sqrt{11}]$ is norm-euclidean