In Problems in Probability by Shiryaev, Many problems about characteristic function are based on these inequalities? Does anybody have hints to prove some of them? Here's the image. Thank you.
Prove that any characteristic function $\varphi=\varphi(t)$ must satisfy the following inequalities:
$$1-\operatorname{Re} \varphi(n t) \leq n\left[1-(\operatorname{Re} \varphi(t))^{n}\right] \leq n^{2}[1-\operatorname{Re} \varphi(t)], n=0,1,2, \ldots ;\tag{*}$$
$$|\operatorname{Im} \varphi(t)|^{2} \leq \frac{1}{2}[1-\operatorname{Re} \varphi(2 t)] ; \quad 1-\operatorname{Re} \varphi(2 t) \geq 2(\operatorname{Re} \varphi(t))^{2}$$
$$|\varphi(t)-\varphi(s)|^{2} \leq 4 \varphi(0)|1-\varphi(t-s)| ; \quad 1-|\varphi(2 t)|^{2} \leq 4\left[1-|\varphi(t)|^{2}\right]$$
$$|\varphi(t)-\varphi(s)|^{2} \leq 2[1-\operatorname{Re} \varphi(t-s)]$$
$$\frac{1}{2 h} \int_{t-h}^{t+h} \varphi(u) d u \leq(1+\operatorname{Re} \varphi(h))^{1 / 2}, t>0$$
Here is solution to the easiest one of them all:
$\operatorname{Im}(\phi)=\int\sin tx\mu(dx)$
Thus, by Jensen's inequality
$\Big|\operatorname{Im}(\phi)\Big|^2\leq\int\sin^2 tx\mu(dx)=\int \frac{1-\cos 2tx}{2}\,\mu(dx)=\frac{1}{2}(1-\int\cos 2tx\,\mu(dx))-\frac{1}{2}(1-\operatorname{Re}(\phi(2t))$
The otherwise are a little more challenging but require tricks like the one above. Also, notice that
$$\Big|\int\cos tx\,\mu(dx)\Big|\leq1$$
Using this, for instance also gives you the RHS of the first inequality: $$n\Big(1-(\operatorname{Re}(\phi(t))^n\big)=n\big(1-\operatorname{Re}(\phi(t))\big)\big(1+\operatorname{Re}(\phi(t))+\ldots+\operatorname{Re}(\phi(t))^{n-1}\big)\leq n^2\big(1-\operatorname{Re}(\phi(t))\big) $$ The right hand side of the same inequality suggest to see whether this holds: $$ 1-\cos n\alpha \leq n(1-\cos^n(\alpha))$$ for all $\alpha$.