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 Turán-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 Turán-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.