A Double Inequality for the Trigamma Function and Its Applications. (16th July 2014) - Yang, Zhen-Hang; Chu, Yu-Ming; Tao, Xiao-Jing
Abstract : We prove thatp = 1 andq = 2 are the best possible parameters in the interval( 0, ∞ ) such that the double inequalitye p / x + 1 - e - p / x / 2 p < ψ ′ x + 1 < e q / x + 1 - e - q / x / 2 q holds forx > 0 . As applications, some new approximation algorithms for the circumference ratioπ and Catalan constantG = ∑ n = 0 ∞ - 1 n / ( 2 n + 1 ) 2 are given. Here, ψ ′ is the trigamma function.