  In the present note, an alternative proof is supplied for Theorem~1 in [N.
Elezovi\'c, C. Giordano and J. Pe\v{c}ari\'c, \textit{The best bounds in
Gautschi's inequality}, Math. Inequal. Appl. \textbf{3} (2000),
239\nobreakdash--252.].
