Logarithmically completely monotonic functions concerning gamma and digamma functions

Abstract

For given real numbers a≥0, b∈ℝ and c∈ℝ, let F a, b, c (x)=[Γ(x+1)]1/x (1+a/x) x+b /x c and φ a, b, c (x)=ψ″(x)+[2+(b+c)x−2x 2]/x 3+[3a(2a−b)+(6a−b)x+2x 2]/(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 F a, b, c (x) for 2a≤3b≤−3c and its reciprocal 1/F a, b, c (x) for 2a≤3b and 1+2b+c≥0 are logarithmically completely monotonic in (0, ∞), while the function φ a, b, c (x) for 0≤2a≤3b and 1+2b+c≥0 and its negative−φ a, b, c (x) for 0≤2a≤3b and b+c≤0 are completely monotonic in (0, ∞).