Inequality on characteristic functions (probability theory)

Question

Show that for every real characteristic function $\phi(t)$ we have

$$1-\phi(2t) \le 4(1-\phi(t))$$

I am not sure where to begin. It seems I am missing some formula or theorem, or is it really that I have to write it out by definition and integrate?

Answer 1

Integrate with respect to the distribution of the random variable the inequality $$1-\cos(2tx)=2-2\cos^2(tx)=2(1+\cos(tx))(1-\cos(tx))\leqslant4(1-\cos(tx)).$$