Using Jensen's inequality to prove the Cauchy distribution has no mean

Question

I can see that there is no mean because $\int x / \pi(1+x^{2})$ does not converge from -inf to inf. But my prof hinted at using Jensen's inequality for the proof.

$$f(E(X)) \le E(f(X))$$

How can I use this? We have not learned Cauchy principal value, so that is not applicable.