Some classes of completely monotonic functions. II (Q850531)

[For part I see Ann. Acad. Sci. Fenn., Math. 27, No. 2, 445--460 (2002; Zbl 1021.26002).] A function \(f:(0,\infty )\rightarrow \mathbb{R}\) is said to be completely monotonic if \((-1)^{n}f^{(n)}(x)\geq 0\) for all \(x>0\) and \(n=0,1,2,\dots\) The authors present some new classes of completely monotonic functions. They are defined in terms of the classical gamma function \(\Gamma \) and their logarithmic derivatives (the digamma and polygamma functions). For example, they investigate the complete monotonicity of the functions defined by \[ g_{a}(x)=x^{a}\left( \Gamma (1+1/x)\right) ^{x}, \qquad H_{a,b}(x)=\left[ x^{a}(e/x)^{x}\Gamma (x)\right] ^{b}, \] \[ Q_{a,b}(x)=\left[ \Gamma (x+a+1) \right] ^{1/(x+a)}/\left[ \Gamma (x+b+1)\right] ^{1/(x+b)}, \quad f_{a,b,\alpha ,\beta }(x)=\left[ \Gamma (ax)\right] ^{\alpha }/\left[ \Gamma (bx)\right] ^{\beta }, \] for \(a,b,\alpha ,\beta \in \mathbb{R}.\)