ML inequality for $ \int \frac{1}{z^2-1} \, dz $

Question

Prove, using ML inequality (estimation lemma), that $$\left |\int f(z) \, dz\right| < \frac{π}3 \,,$$ where $$f(z) = \frac{1}{z^2-1}\,,$$ over the arc of the circle $|z|=2$ that lies in the first quadrant.

So the length of the contour is π, but I can't get a suitable upper bound for $|f(z)|$. After factorising the denominator and using reverse triangle inequality for both fractions, I just get $$\left|\frac{1}{z-1}\right| \le 1 $$ and $$\left|\frac{1}{z+1}\right| \le 1 $$, what am I doing wrong? Thanks!

Answer 1

Using $|z_1-z_2|\ge ||z_1|-|z_2||$ (See This Article), we have on $|z|=2$

$$\left|\frac1{z^2-1}\right|\le \frac{1}{\left||z|^2-1\right|}=\frac1{3}$$

Can you finish now?