Logarithmically completely monotonic functions concerning gamma and digamma functions.

For given real numbers a≥0, b∈ and c∈, let Fa, b, c(x)=[Γ(x+1)]1/x(1+a/x)x+b/xc and φa, b, c(x)=ψ''(x)+[2+(b+c)x-2x2]/x3+[3a(2a-b)+(6a-b)x+2x2]/(x+a)3 with x∈(0, ∞), where Γ(x) and ψ(x) are the well-known Euler gamma function and the psi or digamma function, respectively. In this article, it is revealed that the function Fa, b, c(x) for 2a≤3b≤-3c and its reciprocal 1/Fa, b, c(x) for 2a≤3b and 1+2b+c≥0 are logarithmically completely monotonic in (0, ∞), while the function φa, b, c(x) for 0≤2a≤3b and 1+2b+c≥0 and its negative-φa, b, c(x) for 0≤2a≤3b and b+c≤0 are completely monotonic in (0, ∞). [ABSTRACT FROM AUTHOR]