On van de Lune-Alzer's inequality

In this paper it is shown that the inequality known in the literature as Alzer’s inequality (1993), has already been known since 1975. and is due to Jan van de Lune. A review of different methods in proving Van de Lune - Alzer’s inequality and generalizations in a several directions, is given. It is shown how some results and proofs can be corrected, refined and extended. New results, inspired by the generalization of Van de Lune - Alzer’s inequality for increasing convex
sequences presented by N. Elezovic and J. Pe ´ cari ˇ c, are obtained.