Andrea Ivo Antonio Laforgia

Some Inequalities for Modified Bessel Functions

We denote by and the Bessel functions of the first and third kinds, respectively. Motivated by the relevance of the function , , in many contexts of applied mathematics and, in particular, in some elasticity problems Simpson and Spector (1984), we establish new inequalities for . The results are based on the recurrence relations for and and the Turán-type inequalities for such functions. Similar investigations are developed to establish new inequalities for .

On some Turán-type inequalities

We prove Turán-type inequalities for some special functions by using a generalization of the Schwarz inequality.

Supplements to known monotonicity results and inequalities for the gamma and incomplete gamma functions

We denote by and the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to, a lower bound established by Elbert and Laforgia (2000) for the function, with, only for.

A theory of divergent integrals

Abstract We construct a theory of divergent integrals, based on the following definition: we say that f is ( c , α ) integrable, α > − 1 , if, for some a ∈ R , lim T → ∞ ∫ 0 T ( 1 − t T ) α f ( t ) d t = a . As a consequence of this definition, some theorems are proved. We study, in detail, the case α = 1 .

Van der Corput inequalities for Bessel functions

In this note, we offer some log-concavity properties of certain functions related to Bessel functions of the first kind and modified Bessel functions of the first and second kinds, by solving partially a recent conjecture on the log-convexity/log-concavity properties for modified Bessel functions of the first kind and their derivatives. Moreover, we give an application of the mentioned results by extending two inequalities of van der Corput to Bessel and modified Bessel functions of the first kind. Similar inequalities are proved also for modified Bessel functions of the second kind, as well as for log-concave probability density functions.