zbMATH — the first resource for mathematics

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.

33B15 Gamma, beta and polygamma functions
26D15 Inequalities for sums, series and integrals
26D07 Inequalities involving other types of functions
Full Text: DOI EuDML
[1] Csordas, G; Norfolk, TS; Varga, RS, The Riemann hypothesis and the Turán inequalities, Transactions of the American Mathematical Society, 296, 521-541, (1986) · Zbl 0602.30030
[2] Elbert, Á; Laforgia, A, Some monotonicity properties of the zeros of ultraspherical polynomials, Acta Mathematica Hungarica, 48, 155-159, (1986) · Zbl 0611.33011
[3] Elbert, Á; Laforgia, A, Monotonicity results on the zeros of generalized Laguerre polynomials, Journal of Approximation Theory, 51, 168-174, (1987) · Zbl 0645.33013
[4] Gradshteyn IS, Ryzhik IM: Table of Integrals, Series, and Products. 6th edition. Academic Press, California; 2000:xlvii+1163. · Zbl 0981.65001
[5] Katkova OM: Multiple positivity and the Riemann zeta-function.http://arxiv.org/abs/math.CV/0505174 · Zbl 0512.34018
[6] Laforgia, A, Sturm theory for certain classes of Sturm-Liouville equations and turánians and Wronskians for the zeros of derivative of Bessel functions, Indagationes Mathematicae, 44, 295-301, (1982) · Zbl 0512.34018
[7] Laforgia A, Natalini P: Turán-type inequalities for some special functions. submitted submitted · Zbl 1126.26017
[8] Lorch, L, Turánians and Wronskians for the zeros of Bessel functions, SIAM Journal on Mathematical Analysis, 11, 223-227, (1980) · Zbl 0446.33011
[9] Pólya G: Collected Papers. Vol. II: Location of Zeros, edited by R. P. Boas, Mathematicians of Our Time. Volume 8. The MIT Press, Massachusetts; 1974.
[10] Szegö G: Orthogonal Polynomials, Colloquium Publications. Volume 23. 4th edition. American Mathematical Society, Rhode Island; 1975:xiii+432.
[11] Titchmarsh EC: The Theory of the Riemann Zeta-Function. The Clarendon Press, Oxford; 1951:vi+346. · Zbl 0042.07901
[12] Turán, P, On the zeros of the polynomials of Legendre, Časopis Pro Pěstování Matematiky, 75, 113-122, (1950) · Zbl 0040.32303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.