Recently, several authors have studied the monotonicity properties of functions such as \(\psi(x)= \Gamma'(x)/\Gamma(x)\,.\) See, for instance, [\textit{H. Alzer, C. Berg}, ``Some classes of completely monotonic functions. II.'' Ramanujan J. 11, 225--248 (2006; Zbl 1110.26015), \textit{N. Batir}, J. Math. Anal. Appl. 328, No.~1, 452--465 (2007; Zbl 1105.33002), \textit{C.-P. Chen, F. Qi} J. Math. Anal. Appl. 321, No.~1, 405--411 (2006; Zbl 1099.33002), \textit{S. Guo, F. Qi} and \textit{H. M. Srivastava}, ``Supplements to a class of logarithmically complete monotonic functions associated with the gamma function'', Appl. Math. Comput. (to appear) doi:10.1016/j.amc.2007.08.011 In the paper under review, the author refines three results of \textit{S.-L. Qiu} and \textit{M. Vuorinen} [Math. Comput. 74, No.~250, 723--742 (2005; Zbl 1060.33006)].