Inequalities for characteristic functions involving Fisher information (Q875319)

Let \(f(x)\) be a one-dimensional probability density with characteristic function \(\phi(t)\). The author gives several estimates from above for \(\text{Re } \phi(t)\), \(\text{Im } \phi(t)\) and \(| \phi(t)| ^2\) in terms of the Fisher information \(I(f):=\int ((d/dx)\ln f(x))^2 f(x)\, dx\). In particular, the estimate \(| \phi(t)| ^2\leq I(f)/(I(f)+t^2)\), \(t\in R\), holds.