A Double Inequality for the Trigamma Function and Its Applications

Abstract

We prove that p=1 and q=2 are the best possible parameters in the interval (0,∞) such that the double inequality ep/x+1-e-p/x/2p0. As applications, some new approximation algorithms for the circumference ratio π and Catalan constant G=∑n=0∞-1n/(2n+1)2 are given. Here, ψ′ is the trigamma function.