The authors consider a function introduced by \textit{G. D. Anderson}, \textit{S.-L. Qiu}, \textit{M. K. Vamanamurthu} and \textit{M. Vuorinen} [Pac. J. Math. 192, No. 1, 1-37 (2000; Zbl 0951.33012)], viz., \[ \begin{multlined} {\mathcal L}(a,b,c,r)= {_2F_1}[a- 1,b; c;r]{_2F_1}[a, b;c; 1-r]+ {_2F_1}[a- 1,b; c;1-r]{_2F_1}[a, b;c,r]-\\ {_2F_1}[a, b;c,r]{_2F_1}[a, b;c;10r],\qquad r\in (0,1),\end{multlined} \] where \(a\), \(b\), \(c\) are positive. Their main result is Theorem 2.1: (1) when certain inequalities are satisfied, \({\mathcal L}\) is strictly convex [concave], (2) \({\mathcal L}\gtreqless 0\) according as \(C\gtreqless b\), (3) \({\mathcal L}\) is constant for \(a+ b=1\), (4) \({\mathcal L}\) has precisely one extremum, namely for \(r={1\over 2}\). The proof is based upon Gauss' relations between contiguous functions, and the desired expressions for the derivatives \({\mathcal L}'\) and \({\mathcal L}''\) are obtained by lengthy series manipulations. As corollarties some equations and inequalities involving \({_2F_1}\) are given. An example is Corollary 3.2, \[ {{_2F_1}[a, b;c;r]\over {_2F_1}[a, b; c;1-r]}< {{_2F_1}[a, b;c+ 1;r\over {_2F_2}[a, b; c+1;1-r]},\qquad r\in (0,\textstyle{{1\over 2}}); \] the sign is reversed for \(r\in({1\over 2}, 1)\). \{Note that Corollary 3.1 may be generalized; see SIAM Review Problem 77-2\}.