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The classical inequalities of Bochkarev play a very important role in harmonic analysis. The meaning of these inequalities lies in the connection between the metric characteristics of functions and the summability of their Fourier coefficients. One of the most important directions of harmonic analysis is the theory of Fourier series. His interest in this direction is explained by his applications in various departments of modern mathematics and applied sciences, as well as the availability of many unsolved problems. One of these problems is the study of the interrelationships of the integral properties of functions and the properties of the sum of its coefficients. The solution of these problems was dedicated to the efforts of many mathematicians. And further research in this area are important and interesting problems and can give new, unexpected effects. In the article we receive a two-dimensional analog of the Bochkarev type theorem for the Fourier transform.

Transversal displacements of metal in the hearth of deformation reduce unevenness of elongations across strip width, wherein the camber curvature becomes smaller, and its radius is larger than that calculated on the assumption of plane deformation scheme. The investigations are based on the variational principle of possible changes of strain state (Jourdain principle). It is assumed that the input velocity unevenness of the metal flow in the hearth of deformation across the strip width passes into an output one proportionally to reductions. Further, velocities and powers of deformation and displacements of metal in the hearth of deformation during hot rolling and corresponding powers are calculated. Solving Euler-Poisson's equation, we receive the expression for coefficients, which take into account the influence of the transversal displacements of metal on decreasing the output unevenness and the radius of camber curvature for every harmonic. [ABSTRACT FROM AUTHOR]

The question of the global solvability of a mixed problem for a Schrödinger evolution equation with a cubic dissipative term is considered. [ABSTRACT FROM AUTHOR]

Abstract: Some generalizations of the well-known Cauchy–Schwarz inequality and the analogous Cauchy–Bunyakovsky inequality involving four free parameters are given for both discrete and continuous cases. Several particular cases of interest are also analyzed. Some of the applications of our main results include (for example) the Wagner inequality. [Copyright &y& Elsevier]