On the absolute monotonicity of generalized elliptic integral of the first kind

It was proved by Wang et al. (Appl Anal Discrete Math 14:255–271, 2020) that twice derivative of the function x↦K(x)-log(1+4/1-x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\mapsto {{\,\mathrm{\textit{K}}\,}}(\sqrt{x})-\log (1+4/\sqrt{1-x})$$\end{document} is absolutely monotonic on (0, 1). This will be, in this paper, extended to the generalized elliptic integral of the first kind, more precisely, we will study the absolutely monotonic properties of the function x↦Ka(x)-log(1+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}$$x\mapsto {{\,\mathrm{\textit{K}}\,}}_a(\sqrt{x})-\log \Big (1+c/\sqrt{1-x}\Big )$$\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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{\textit{K}}\,}}_{a}$$\end{document} is the generalized elliptic integral of the first kind.