A note on a function involving complete elliptic integrals: monotonicity, convexity, inequalitiesЗаметка об одной функции, связанной с полными эллиптическими интегралами: монотонность, выпуклость, неравенства

We study monotonicity and convexity properties of the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Delta (r) = \frac{{\varepsilon - r'^2 \mathcal{K}}} {{r^2 }} - \frac{{\varepsilon ' - r^2 \mathcal{K}'}} {{r'^2 }}$\end{document}, where κ and ε denote the complete elliptic integrals of the first and second kind, respectively. Moreover, we present some inequalities for Δ.