Inequalities of the Turán-Type for the Le Roy Type's Mittag-Leffler Function

Abstract

This paper presents the necessary and sufficient conditions for monotonicity of the Mittag-Leffler function of the Le Roy type (abbr. MLR-functions), taking into account its special place in the theory of analytic functions. Mehrez and Sitnik studied the monotonicity of the ratio of sections on the series of Mittag-Leffler function (MLF) and developed some Tur & aacute;n-type inequalities. These inequalities have a wide range of applications in understanding the analytical properties of functions. The proposed work is a study for MLR-functions. The primary objective of this study is the development of the Tur & aacute;n-type inequalities and subsequent validation of these inequalities on specific intervals. In addition, the log-convex property of this function is analyzed, and the theoretical significance of this property is elaborated.