On evolution of the integral of the product of two real functions with Levin–Stechkin type of inequality

Classical integral Levin–Stechkin inequality known since 1948. This article received more complex and accurate inequality, with which you can get the upper or lower bounds of the integral of the product of two real functions,using the generalized Fourier coefficients of each of these functions separately. Fourier coefficients calculated according to a certain Chebyshev set of functions and it is assumed that the real valued functions added to the specified set also form Chebyshev systems. It is suggested also that all functions in Chebyshev systems are mutually orthogonal, as it is customary in the proof of Levin–Stechkin inequality. The result is formulated with theorem, which being illustrated with five examples. In two examples Chebyshev systems are orthogonal polynomials on finite intervals, in following two examples the cited functions have specific form and it is shown that increasing the number of Fourier coefficients you can enclose the original integral in narrowing the plug from both the bottom and top boundaries. Last example shows how to use the theorem in the problem of estimation of the variance of the number of zeros of Gaussian stationary process. Bibliogr. 9.