Sharp Approximations for Complete p-Elliptic Integral of the Second Kind by Weighted Power Means

In this paper, the well-known double inequality for the complete elliptic integral E(r) of the second kind, which gives sharp approximations of E(r) by power means (or Hölder means), is extended to the complete p-elliptic integral Ep(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_p(r)$$\end{document} of the second kind, and thus sharp approximations of Ep(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_p(r)$$\end{document} by weighted power means are obtained. This result confirmed the truth of Conjecture I by Barnard, Ricards and Tiedeman in the case when a=b=1/p∈(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}$$a=b=1/p\in (0,1/2)$$\end{document} and c=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=1$$\end{document} and also provides a new method to prove the above double inequality of E(r).