Sharp Lyapunov's inequality for the measurable sets with infinite measure, with generalization to the Grand Lebesgue spaces

Abstract: We extend the classical Lyapunov inequality on the measurable space with infinite measure and on the so-called Grand Lebesgue spaces (GLS). We find also the exact value for correspondent constant. Possible applications: Functional Analysis (for instance, interpolation of operators), Integral Equations, Probability Theory and Statistics (tail estimations for random variables) etc.