DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

MONOTONICITY RESULTS AND INEQUALITIES
FOR SOME SPECIAL FUNCTIONS
A. LAFORGIA AND P. NATALINI
By using a generalization of the Schwarz inequality we prove, in the first part of this
paper, Turán-type inequalities relevant to some special functions as the psi-function, the
Riemann ξ-function, and the modified Bessel functions of the third kind. In the second
part, we prove some monotonicity results for the gamma function.
Copyright © 2006 A. Laforgia and P. Natalini. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In the first part of this paper, we prove new inequalities of the following type:
f n (x) f n+2 (x) − f 2
n+1(x) ≤ 0, (1.1)
with n = 0,1,2,..., which have importance in many fields of mathematics. They are
named, by Karlin and Szegö, Turánians because the first type of inequalities was proved
by Turán [13]. More precisely, by using the classical recurrence relation [11, page 81]
(n +1)P n+1 (x) = (2n +1)xP n (x) − nP n−1 (x), n = 0,1,...,
P −1 (x) = 0, P 0 (x) = 1,
(1.2)
and the differential relation [11, page 83]
( 1 − x
2 ) P ′ n(x) = nP n−1 (x) − nxP n (x), (1.3)
he proved the following inequality:
P n (x) P n+1 (x)
≤ 0,
∣P n+1 (x) P n+2 (x) ∣
−1 ≤ x ≤ 1, (1.4)
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 615–621