Several completely monotone functions related to DeTemple's sequence

In this paper, we present the necessary and sufficient conditions such that several functions involving $R\left( x\right) =\psi \left( x+1/2\right) -\ln x$ with a parameter are completely monotone on $\left( 0,\infty \right)$, where $\psi$ is the digamma function. This generalizes some known results and verifies a conjecture posed by Chen.