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On some Turán-type inequalities. (English) Zbl 1095.33002
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.

##### MSC:
 33B15 Gamma, beta and polygamma functions 26D15 Inequalities for sums, series and integrals 26D07 Inequalities involving other types of functions
##### Keywords:
Schwarz inequality; Turán-type inequalities
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##### References:
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