Logarithmically complete monotonicity properties and characterizations of the gamma function

A Property of Logarithmically Absolutely Monotonic Functions and the Logarithmically Complete Monotonicity of a Power-exponential Function

In the article, a notion “logarithmically absolutely monotonic function” is introduced, an inclusion that a logarithmically absolutely monotonic function is also absolutely monotonic is revealed, the logarithmically complete monotonicity and the logarithmically absolute monotonicity of the function ` 1+ α x ́x+β are proved, where α and β are given real parameters, a new proof for the inclusion t...

A Function Involving Gamma Function and Having Logarithmically Absolute Convexity

In the paper, the logarithmically complete monotonicity, logarithmically absolute monotonicity and logarithmically absolute convexity of the function [Γ(1+tx)] [Γ(1+sx)]t for x, s, t ∈ R such that 1 + sx > 0 and 1 + tx > 0 with s 6= t are verified, some known results are generalized.

Monotonicity and Convexity for the Gamma Function

Let a and b be given real numbers with 0 ≤ a < b < a + 1. Then the function θa,b(x) = [Γ(x + b)/Γ(x + a)]1/(b−a) − x is strictly convex and decreasing on (−a,∞) with θa,b(∞) = a+b−1 2 and θa,b(−a) = a, where Γ denotes the Euler’s gamma function.

Monotonicity and inequalities for the gamma function

Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function

In the paper, the authors establish integral representations of some functions related to the remainder of Burnside’s formula for the gamma function and find the (logarithmically) complete monotonicity of these and related functions. These results extend and generalize some known conclusions. 1. Motivation and main results A function f is said to be completely monotonic on an interval I if f ha...