Describe all meromorphic functions f(z) in the complex plane with a simple pole at z=1, a simple zero at z=-1, and for which
$$|f(z)|\le M|z|,$$
for $|z|\ge 2$
for some $M>0$.
I know that, since f has a simple pole at z=1, then f(z) must be of the form $$\frac{g(z)}{(z-1)}$$, where g(z) is analytic and non-zero at z=1.
Similarly, since f has a simple zero at z=-1, then f(z) must be of the form $$(z+1)h(z)$$, where h(z) is analytic and non-zero at z=-1.
Combining the two, I have that f(z) must be of the form $$\frac{(z+1)}{(z-1)}w(z)$$, where w(z) is analytic and non-zero at both z=1 and at z=-1.
Using the inequality given, I have that $$|\frac{(z+1)}{(z-1)}w(z)| \le M|z|$$, for $|z|\ge 2$.
How can I proceed from here? ...Or have I started off incorrectly already? I know I haven't said much yet, but, so far, I've used everything that's given in the problem, I think. Also, I wasn't able to derive any new information from moving around some parts in the inequality.
Thanks in advance,
Edit: Perhaps the upper bound is telling me that my function f(z) grows like a polynomial, hence it is a polynomial, but that can't be true, since I am given that f(z) has a simple pole at z=1.
This was sorted out in comments, but here's a summary: by assumptions about the pole and the zero, $f$ has the form $$f(z) = \frac{z+1}{z-1}w(z)\tag{1}$$ where $w$ is an entire function. For $|z|\ge 2$ we have $$ |w(z)|=\frac{|z-1|}{|z+1|}|f(z)| \le \frac{|z|+1}{|z|-1}M|z| = \frac{1+1/|z| }{1-1/|z| }M|z|\le 3M|z| $$ According to Entire function bounded by a polynomial is a polynomial, $w$ is a polynomial of degree at most $1$. And conversely, for any such polynomial $(1)$ gives a function with the required properties.