A Property of Logarithmically Absolutely Monotonic Functions and the Logarithmically Complete Monotonicity of a Power-exponential Function
نویسنده
چکیده
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 that a logarithmically completely monotonic function is also completely monotonic is given, and an open problem is posed.
منابع مشابه
Two Logarithmically Completely Monotonic Functions Connected with Gamma Function
In this paper, the logarithmically complete monotonicity results of the functions [Γ(1 + x)]y/Γ(1 + xy) and Γ(1 + y)[Γ(1 + x)]y/Γ(1 + xy) are established.
متن کاملNecessary and Sufficient Conditions for a Function Involving a Ratio of Gamma Functions to Be Logarithmically Completely Monotonic
In the paper, necessary and sufficient conditions are presented for a function involving a ratio of two gamma functions to be logarithmically completely monotonic, and some known monotonicity and inequality results are generalized and extended.
متن کاملStieltjes-Pick-Bernstein-Schoenberg and their connection to complete monotonicity
This paper is mainly a survey of published results. We recall the definition of positive definite and (conditionally) negative definite functions on abelian semigroups with involution, and we consider three main examples: Rk, [0,∞[k, N0–the first with the inverse involution and the two others with the identical involution. Schoenberg’s theorem explains the possibility of constructing rotation i...
متن کاملIntegral 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...
متن کاملNecessary and Sufficient Conditions for Functions Involving a Ratio of Gamma Functions to Be Logarithmically Completely Monotonic
In the paper, necessary and sufficient conditions are presented for a function and its reciprocal involving a ratio of two gamma functions to be logarithmically completely monotonic, and some known monotonicity and inequality results are generalized and extended.
متن کامل