By employing the convolution theorem of Laplace transforms,
some asymptotic formulas and integral representations of the gamma, psi and
polygamma functions, and other analytic techniques, this note provides an
alternative proof of a monotonicity and convexity property by N. Elezović,
C. Giordano and J. Pečarić in [4] to establish the best bounds in Gautschi-
Kershaw inequalities. Moreover, some (logarithmically) complete monotonicity
results on functions related to Gautschi-Kershaw inequalities are remarked.