An inequality for a multidimensional characteristic function

Let $\xi $ be a vector-valued random variable in $\mathbf{R}^s$ and a corresponding density function $p_\xi(x)$ be “close” to the “standard”normal density. Under this condition an inequality for a characteristic function is proved. The inequality obtained is of interest for the problem of a lower estimator of the rate of convergence in the local limit theorem for densities. An analogous inequality for a lattice distribution was investigated in [N. G. Gamkrelidze, Litovsk. Mat. Sb., 7 (1967), pp. 405–408 (in Russian)] and was given in [V. V. Petrov, Sums of Independent Random Variables, Springer-Verlag, Berlin, New York, 1975] and [Yu. V. Prohorov and Yu. A. Rozanov, Probability Theory: Basic Concepts, Limit Theorems, and Random Processes, Springer-Verlag, Berlin, New York, 1969].