On the Convexity and Concavity of Generalized Complete Elliptic Integral of the First Kind

In this paper, we study the convexity (concavity) of the function x↦Ka(x)-log1+c/1-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\mapsto {{\,\mathrm{{{{\textsf {\textit{K}}}}}}\,}}_a(\sqrt{x})-\log \left( 1+c/\sqrt{1-x}\right) $$\end{document} on (0, 1) for a∈(0,1/2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in (0,1/2]$$\end{document} and c∈(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c\in (0,\infty )$$\end{document}, where Ka(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{{{{\textsf {\textit{K}}}}}}\,}}_a(r)$$\end{document} is the generalized complete elliptic integral of the first kind. This work is an extension of Yang and Tian (Appl Anal Discrete Math 13:240–260, 2019), and also gives a refinement of inequality (Yang and Tian 2019, 0.27) as an application.