Two-sided inequalities for the Struve and Lommel functions

Abstract

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial differential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function ₁F₂(a; b; c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by rst investigating inequalities for the general hypergeometric function ₁F₂(a; b; c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modied Lommel function and the normalized Lommel function.

Mathematics Subject Classication (2010): Primary 26D15, 30A10; Secondary 26D07,
33C20.

Key words: Generalized hypergeometric function pFq, gamma function, Pochhammer
symbol, gamma function, Mellin-Barnes contour integral, con uent hypergeometric func-
tions, Struve functions, associated and modied Struve functions, Lommel functions,
modied and normalized Lommel functions.