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

Abstract

Abstract We prove that the function F α,β (x) = x α Γ 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}. 2010 Mathematics Subject Classification: 33B15; 26A48.

Keywords

• completely monotonic
• logarithmically completely monotonic
• gamma function