On some Turán-type inequalities (Q2491570)

In this paper the inequalities of the type \(f_n(x)f_{n+2}(x)\geq f_{n+1}^2(x),\quad n=0,1,2,\ldots,\) ( called by the authors Turán-type inequalities ) are obtained for some special functions. They are proved by using the following generalization of the Schwarz inequality \[ \int_{a}^{b}g(t)[f(t)]^m \,dt \cdot \int_{a}^{b}g(t)[f(t)]^n \,dt\geq \biggl(\int_{a}^{b}g(t)[f(t)]^{(m+n)/2} \,dt\biggr)^2, \] where \(f, g\) are nonnegative functions of a real variable and \(m, n\) are real numbers, such that the integrals exist.