Finding Possible Meromorphic functions on $\mathbb{C}$

Question

I am trying to find all meromorphic functions on $\mathbb{C}$ such that:

$$ \mid f(z) \mid \leq (\frac{3 \mid z \mid}{\mid z + 1 \mid})^{3/2}$$

Can I express the functions as:

$$f(z) = \frac{C_{-2}}{(z+1)^2}+\frac{C_{-1}}{(z+1)}+C_{0}+C_{1}(z+1)$$

Simply by using the fact that $\frac{3}{2} < 2$ and that the functions are meromorphic or am I overlooking something? I know I still need to figure out restraints on the coefficients.

Answer 1

Your function's only possible pole is at $-1$, and it has to be a simple pole: a pole of order $p$ would have $|f(z)| \sim |z+1|^{-p}$ as $z \to -1$, and this is greater than your right side if $p > 3/2$. Your function must have a removable singularity at $\infty$, because it is bounded as $z \to \infty$. After removing the pole you'll be left with a bounded entire function, which is constant. So it must be of the form

$$ f(z) = \dfrac{C_{-1}}{z+1} + C_0$$

Answer 2

Since $3/2<2$, $C_{-2}$ must be $0$. Except for that, your work is correct. Also, there is no need to write $C_1(z+1)$; you can write $C_1\,z$.