A logarithmically completely monotonic function involving the ratio of gamma functions

In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient conditions for a new function involving the ratio of two gamma functions and originating from the coding gain to be logarithmically completely monotonic.

In [34,35], the function was verified to be logarithmically completely monotonic on (0, ∞).This result was promptly strengthened in [3] to the function (1.5) being a Stieltjes transform.A Stieltjes transform is a function f : (0, ∞) → [0, ∞) which can be written in the form where a, b are nonnegative constants and µ is a nonnegative measure on (0, ∞) such that More generally, the inclusions were discovered in [3,8,27,30], where S, L[(0, ∞)], and C[(0, ∞)] denote respectively the set of all Stieltjes transforms, the set of all logarithmically completely monotonic functions on (0, ∞), and the set of all completely monotonic functions on (0, ∞).An infinitely differentiable function f is said to be completely monotonic on an interval I if it satisfies for all i ∈ {0} ∪ N on I.For more information on the inclusions in (1.7), please rfer to [44] and other references cited therein.Some properties of the function [Γ(x + 1)] 1/x and its logarithm can be found in [5,30,33] and tightly related references therein.
In [36,38], the monotonicity of the function for x, s, t ∈ R such that 1 + sx > 0 and 1 + tx > 0 with s = t, and its general form for ax ∈ I and bx ∈ I, where a and b are two real numbers and f (x) is a positive function on an interval I, are investigated.For a much complete survey of this topic, please read [20, pp.73-76, Section 7.6].
(1.13) Some inequalities and and necessary and sufficient conditions for the function h α,y (x) and its special cases to be logarithmically completely monotonic were provided in [9,10,29,43,45,46] and many other references listed therein.Let s and t be two real numbers and α = min{s, t}.
, s = t, e ψ(x+s) , s = t. (1.14) There have been a large amount of literature devoted to inequalities, logarithmically complete monotonicity, asymptotic expansions, applications, and the like, of functions involving the function (1.14).The work in this field dating back to 1948 has been surveyed and reviewed in [20,39].Recently, some new results on inequalities and logarithmically complete monotonicity of functions involving (1.14) were obtained in [6,26,31,40].We may classify recently published papers relating to the gamma and polygamma functions and to the logarithmically complete monotonicity into groups as follows: (1) Some papers having something to do with the unit ball in R n are [6,11,12,31,43,47].
(2) Some papers on asymptotic expansions for the gamma function Γ or for its ratio such as (1.14) are [4,14,15,16].(3) A series of papers on the complete monotonicity of the function e 1/t − ψ ′ (t) and its variants are [13,18,23,25,42].(4) Some papers on the notion "completely monotonic degree" and its computation are [7,21,23,42].(5) Some papers related to the function [ψ ′ (x)] 2 + ψ ′′ (x) and its divided difference forms are [6,19,21,26,28,31,40,48].(6) Some applications of the complete monotonicity to number theory and mean values are published in [22,44] and some preprints listed there.Now let us return to rearrange the function h(x) in (1.1) as It is not difficult to see that the function h(x) is not a special case of any of the functions in (1.4) and (1.9) to (1.14).On the other hand, the function h(x) may be written in a general form where a, b, c > 0 and x ∈ (0, ∞).The aim of this paper is to find sufficient conditions on a, b, c such that the function h a,b;c (x) or its reciprocal is logarithmically completely monotonic on (0, ∞).
Our main results may be stated as the following theorem.and and Since  Remark 2.2.The techniques in the proof of Theorem 1.1 and [17, Appendix B] were ever appeared in [33].

1 )
for ℜ(z) > 0 and n ∈ N, see[1, p. 260, 6.4.1], the function (−1) k ψ (k) (x) is increasing on (0, ∞) for k ∈ N. Hence, when a > b, the derivative H ′ a,b;c;k (x) is positive on (0, ∞) for k ∈ N.This means that, when a > b, the function H a,b;c;k (x) is increasing on (0, ∞) for k ∈ N. Therefore, when a > b and c ≥ Γ(b) Γ(a) , the function H a,b;c;k (x) is positive on (0, ∞) for k ∈ N. Consequently, when a > b and c ≥ Γ(b) Γ(a) , it follows that (−1) k [ln h a,b;c (x)] (k) = k! x k+1 H a,b;c;k (x) ≥ 0 (2.2) for k ∈ N on (0, ∞), that is, the function h a,b;c (x) defined by (1.16) is logarithmically completely monotonic.Similarly, when a < b, the derivative H ′ a,b;c;k (x) is negative and the function H a,b;c;k (x) is decreasing on (0, ∞) for k ∈ N. Therefore, when a < b and c ≤ Γ(b) Γ(a) , the function H a,b;c;k (x) is negative and the inequality (2.2) is reversed on (0, ∞) for k ∈ N.This implies that the reciprocal of the function h a,b;c (x) defined by (1.16) is logarithmically completely monotonic.The proof of Theorem 1.1 is complete.

Remark 2 . 1 .
It is apparent that the proof of Theorem 1.1 is slightly simpler than the proof in [17, Appendix B] and that Theorem 1.1 generalizes the result obtained in [17, Appendix B].