Sharp inequalities for the generalized elliptic integrals of the first kind

Elliptic integrals are of cardinal importance in mathematical analysis and in the field of applied mathematics. Since they cannot be represented by the elementary transcendental functions, there is a need for sharp computable bounds for the family of integrals. In this paper, by studying the monotonicity of the functions Gpr=p+r2KarlneRa/2/r′andIpr=p+r2Kar-pπ/2ln1/r′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {G}_{p}\left( r\right) =\frac{\left( p+r^{2}\right) \mathcal {K} _{a}\left( r\right) }{\ln \left( e^{R\left( a\right) /2}/r^{\prime }\right) } \quad \text {and }\quad \mathcal {I}_{p}\left( r\right) =\frac{\left( p+r^{2}\right) \mathcal {K}_{a}\left( r\right) -p\pi /2}{\ln \left( 1/r^{\prime }\right) } \end{aligned}$$\end{document}on 0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( 0,1\right) $$\end{document}, we establish some new sharp lower and upper bounds for the generalized elliptic integrals of the first kind Kar\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K} _{a}\left( r\right) $$\end{document}, where Rx≡Rx,1-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R\left( x\right) \equiv R\left( x,1-x\right) $$\end{document} is the Ramanujan constant function defined on (0, 1 / 2], r′=1-r2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r^{\prime }=\sqrt{ 1-r^{2}}$$\end{document}, p∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in \mathbb {R}$$\end{document} is a parameter. These results not only improve some known bounds in the literature, but also yield some new inequalities for Kar\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {K}_{a}\left( r\right) $$\end{document}.