Univalence and convexity conditions for certain integral operators associated with the Lommel function of the first kind

Abstract

A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.

Keywords

• analytic functions
• univalent functions
• integral operators
• special functions
• lommel functions
• univalence
• convexity
• quantum (or q-) calculus