Some new properties of the beta function and Ramanujan R-function

In this paper, the power series and hypergeometric series representations of the beta function and the Ramanujan R-function with one parameter, Bx=Gamma x2 Gamma 2xandRx=-2 psi x-2 gamma,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {B}}\left( x\right) =\frac{\Gamma \left( x\right) <^>{2}}{\Gamma \left( 2x\right) }\text { and }{\mathcal {R}}\left( x\right) =-2\psi \left( x\right) -2\gamma , \end{aligned}$$\end{document}are presented, which yield higher order monotonicity results related to B(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathcal {B}}(x)$$\end{document} and R(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}(x)$$\end{document}; the decreasing property of the functions Rx/Bx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}\left( x\right) /{\mathcal {B}}\left( x\right) $$\end{document} and [B(x)-R(x)]/x2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[ {\mathcal {B}}(x) -{\mathcal {R}}(x)] /x<^>{2}$$\end{document} on 0,infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( 0,\infty \right) $$\end{document} is proved. Moreover, a conjecture put forward by Qiu et al. in [17] is proved to be true. As applications, several inequalities and identities are deduced. These results obtained in this paper may be helpful for the study of certain special functions. Finally, an interesting infinite series similar to the Riemann zeta functions is mentioned and a relevant problem is proposed.