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 $(0,\mathrm{\infty })$ such that the double inequality holds for $x>0$. As applications, some new approximation algorithms for the circumference ratio $\pi$ and Catalan constant $G={\sum }_{n=0}^{\infty }\left({{\left(-1\right)}^n/(2n+1)}^2\right)$ are given. Here, ${\psi }^{\mathrm{\prime }}$ is the trigamma function.