A double inequality for the trigamma function and its applications
Summary: We prove that \(p = 1\) and \(q = 2\) are the best possible parameters in the interval \((0, \infty)\) such that the double inequality \(\left(e^{p / \left(x + 1\right)} - e^{- p / x}\right) / 2 p < \psi' \left(x + 1\right) < \left(e^{q / \left(x + 1\right)} - e^{- q / x}\right) / 2 q\) holds for \(x > 0\). As applications, some new approximation algorithms for the circumference ratio \(\pi\) and Catalan constant \(G = \sum_{n = 0}^\infty \left(\left(- 1\right)^n /(2 n + 1)^2\right)\) are given. Here, \(\psi'\) is the trigamma function.
scientific article; zbMATH DE number 7022908
scientific article