A Double Inequality for the Trigamma Function and Its Applications - Zhen-Hang Yang; Yu-Ming Chu; Xiao-Jing Tao (Abstract and Applied Analysis)
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/2p<ψ′x+1<eq/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=∑n=0∞-1n/(2n+1)2 are given. Here, ψ′ is the trigamma function.