A completely monotonic function involving the gamma and tri-gamma functions

Abstract

In the paper the author provides necessary and sufficient conditions on $a$ for the function $\frac{1}{2}\ln (2\pi )-x+(x-\frac{1}{2})\ln x-\ln \Gamma (x)+\frac{1}{12}{\psi }^{\prime }(x+a)$ and its negative to be completely monotonic on $(0,\infty )$, where $a\ge 0$ is a real number, $\Gamma (x)$ is the classical gamma function, and $\psi (x)=\frac{{\Gamma }^{\prime }(x)}{\Gamma (x)}$ is the di-gamma function. As applications, some known results and new inequalities are derived.

10 pages

References in corpus (8)

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Cited by in corpus (3)

• Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function
• Asymptotic formulas and inequalities for gamma function in terms of tri-gamma function
• Some inequalities for the trigamma function in terms of the digamma function