Define and parameterize a suitable circle $\gamma$ for which the following inequality

Question

If $R> 0$ and $R \ne 0$, we are asked to define and parameterize a suitable circle $\gamma$ for which the following inequality is valid $$\int_\gamma\frac{z+1}{z-1}\mathrm dz\le\frac{2\pi R(r+1)}{R-1}$$

Answer 1

Assuming that you are talking about a circle centred at the origin. Then you can simply use the residue theorem (or Cauchy's integral formula) to find the two possible values the integral can take. This makes it easy to find a general answer.

More simply, you can choose $R<1$ because then, the circle does not enclose the zero at $z=1$ and hence, the integral vanishes.