Remarks on a paper by Chao-Ping Chen and Feng Qi

In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence P n := 1.3 … ( 2 n − 1 ) 2.4 … 2 n P_{n}:=\frac {1.3\ldots (2n-1)}{2.4\ldots 2n} . We show that their result follows easily from a theorem of G. N Watson published in 1959. We also show that the main result of Chen and Qi’s paper is a special case of a more general inequality which admits a very short proof.