On maximum order statistics from heterogeneous geometric variables

Let X1,X2 be independent geometric random variables with parameters p1,p2, respectively, and Y1,Y2 be i.i.d. geometric random variables with common parameter p. It is shown that X2:2, the maximum order statistic from X1,X2, is larger than Y2:2, the second order statistic from Y1,Y2, in terms of the hazard rate order [usual stochastic order] if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\geq \tilde{p}$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde{p}=(p_{1}p_{2})^{\frac{1}{2}}$\end{document} is the geometric mean of (p1,p2). This result answers an open problem proposed recently by Mao and Hu (Probab. Eng. Inf. Sci. 24:245–262, 2010) for the case when n=2.