Inequality related to the real part of a Characteristic function of a random variable.

Question

If $u= Re(\phi),$ for some characteristic function $\phi$, show that $u^2(t)\le \frac{1}{2}(1 + u(2t))$. Hence show that $|\phi(t)|^2 \le \frac{1}{2}(1+|\phi(2t)|)$.

The first part I could show. Stuck with the second part.

Answer 1

$$(\mbox{Re} E(e^{it} ))^2= \left(\mbox{Re} \int_{A} e^{it} dP\right)^2=\left(\mbox{Re} \int_{A} \cos t dP\right)^2\leq \mbox{Re} \int_{A} \cos^2 t dP =\mbox{Re} \int_{A} \frac{1 +\cos 2t}{2} dP=\frac{1}{2} Re(1 +E(e^{i2t} ))$$