  In this paper, the power series and hypergeometric series representations of
the beta and Ramanujan functions \begin{equation*} \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{equation*}
are presented, which yield higher order monotonicity results related to $
\mathcal{B}(x)$ and $\mathcal{R}(x)$; the decreasing property of the functions
$\mathcal{R}\left( x\right) /\mathcal{B}\left( x\right) $ and $[ \mathcal{B}(x)
-\mathcal{R}(x)] /x^{2}$ on $\left( 0,\infty \right)$ are 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 Riemann zeta
functions is observed initially.
