We prove that p = 1 and q = 2 are the best possible parameters in the interval (0,∞) such that the double inequality (e p/(x+1)-e -p/x)/2p < ′ (x + 1) > (e q/(x+1)-e -q/x)/2q holds for x > 0. As applications, some new approximation algorithms for the circumference ratio π and Catalan constant G = Σ n=0∞ ((-1) n/(2n + 1)2) are given. Here, ψ ′ is the trigamma fun...
