Monotonicity and logarithmic concavity of two functions involving exponential function

Complete monotonicity of a function involving the p-psi function and alternative proofs

“…The function ϕ(t) is increasing on (0, ∞) with See [5,6,7,11,14,15,18] and related references therein. Therefore, for x > 0 and n ∈ N, we have…”

Two New Proofs of the Complete Monotonicity of a Function Involving the Psi Function

“…Multiplying on all sides of (35) by x and rearranging gives In [9,12,21] and related references therein, the function h(t) was shown to be decreasing on (−∞, ∞), concave on (−∞, 0) and convex on (0, ∞). This means that the function θ 1 (x) is completely monotonic on (0, ∞) and that the second limit in (16) follows.…”

A logarithmically completely monotonic function involving the gamma function and originating from the Catalan numbers and function

“…Since the function β(t) is positive on (0, ∞), see [3,17,29] and references therein, the right hand side of (3) is a completely monotonic function on (0, ∞). This means that the function…”