A function involving gamma function and having logarithmically absolute convexity

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.

Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions

A logarithmically completely monotonic function involving the ratio of gamma functions

In the paper, the authors first survey and review some logarithmically completely monotonic functions involving the gamma function and its various ratios, and then find sufficient conditions for a function involving the ratio of two gamma functions and originating from the coding gain to be logarithmically completely monotonic.

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.

Absolute monotonicity of a function involving the exponential function

In the paper, the author verifies the absolute monotonicity of a function involving the exponential function.