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We prove that the function F α,β(x) = x αΓ β(x)/Γ(βx) is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) ∈ {(α, β): β > 0, β ≥ 2α + 1, β ≥ α + 1}\{(α, β): α = 0, β = 1} and that [F α,β(x)] -1 is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) ∈ {(α, β): β > 0, β ≤ 2α + 1, β ≤ α + 1}\{(α, β): α = 0, β = 1}. © 2011 Lv et al; licensee Springer.

Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic