Characteristic function inequality

Random variable $X$ and its characteristic function $\phi_X(t)$ then $$\Pr\left(|X|>\frac2T\right) \leq 2\left(1 - \frac1{2T}\int_{-T}^{T}\phi_X(t)dt\right)$$ I cannot find a way how to upperbound or compute integral with characteristic function, also I would be interested, what is interpretation of such integral in terms of probability.

• Replace $\phi_X$ by its definition (as an integral over the real numbers), switch the order of integration, perform the $t$ integral, and apply the obvious bound to the result. – whuber Feb 8 '18 at 19:24