Let $\varphi(t) $ be a characteristic function of some random variable. Proof the following inequality: $$ \left| \frac{1}{2h} \int\limits_{t-h}^{t+h} \varphi(u) d u \right| \le \left(1 + \text{Re}\: \varphi(h) \right)^{1/2}$$
I tried to rewrite it somehow, but I failed all attempts. I will be very greatful if you help!