An estimate for the integral of the modulus for a generalized Poisson-distribution characteristic function over an interval of small length (Q1603209)

The author considers problems of estimation of integrals of the form \(\int_\triangle e^{h(t)}dt,\) where \(\triangle \) is an interval, \(h(t)=(1/2)\int_{-\infty}^{\infty}(\cos tx-1)dH(x)\), \(H(x)\) is a nondecreasing function. New estimates are obtained. Comparison with previous results and some corollaries are given. The function \(e^{h(t)}\) could be considered as the modulus of characteristic function of a certain generalized Poisson distribution and the results could be used for obtaining new limit theorems.