A simple inequality for the variance of the number of zeros of a differentiable Gaussian stationary process

The variance of the number of zeros of a Gaussian differentiable stationary process in a finite time interval can be represented by a single integral of a sophisticated function having singularities in the vicinity of zero, which complicates computer calculations. In this paper, for a wide class of correlation functions, an inequality estimating this variance in simpler terms is proved. Two of three considered examples demonstrate the limits of the effectiveness of the obtained inequality by comparison with special processes earlier established by the author for which the variance is calculated by formulas without integrals. In the two subsequent cases, the inequality is used for the asymptotic estimation of the variance of the number of zeros in a small time interval and, in the last one, in addition to this asymptotics, the upper and lower bounds for the most widely used analytic process in all time intervals.