On Some Complete Monotonicity of Functions Related to Generalized k−Gamma Function

(− 1)L(u)≥ 0, u ∈ I; n ∈ N. (3) Completely monotonic functions have remarkable applications in different fields such as in the theory of special functions, numerical and asymptotic analysis, probability, and physics. Some important properties of these functions were collected in [5], and for more information about this topic, we refer the reader to [6, 7]. In 2007, Alzer and Batir [8] proved that the function


Introduction
e ordinary gamma function is given by the following equation [1]: Γ(u) � lim n⟶∞ n ! n u− 1 u(u + 1)(u + 2), . . . , (u +(n − 1)) , u > 0, (1) which was discovered by Euler when he generalized the factorial function to noninteger values. Lots of mathematicians studied the gamma function because of its great importance; for complete studies of the gamma function, please refer to [2,3]. e digamma function is the logarithmic derivative of the gamma function and is given by [4]: where c � lim n⟶∞ ( n s�1 (1/s) − log n) is Euler-Mascheroni's constant. A function L(x) defined on an interval I is said to be completely monotonic if it possesses derivatives L (n) (u) for all n ∈ N such that (− 1) n L (n) (u) ≥ 0, u ∈ I; n ∈ N. (3) Completely monotonic functions have remarkable applications in different fields such as in the theory of special functions, numerical and asymptotic analysis, probability, and physics. Some important properties of these functions were collected in [5], and for more information about this topic, we refer the reader to [6,7].
In 2007, Alzer and Batir [8] proved that the function is completely monotonic on (0, ∞) if and only if α ≥ (1/3), and they also proved the function − M α (u) is completely monotonic on (0, ∞) if and only if α � 0. As a consequence, they introduced the following sharp bounds for Γ function in terms of ψ function: with the best possible constants c � (1/3) and d � 0. In order to refine inequality (5), Batir [9] modified the function M 0 (u) and proved that the function which is completely monotonic on (0, ∞) if and only if δ ≥ (1/4), and the function − T δ (u) is completely monotonic on (0, ∞) if and only if δ ≤ 0. As a consequence, he deduced the following refinement of the inequality (5): with the best possible constants a � (1/4) and b � 0. In many contexts, such as the combinatorics of creation, the annihilation operators, and the perturbative computation of Feynam integrals, the following symbol appears [10][11][12]: which is a generalization of the ordinary Pochhammer symbol (u) n � ((Γ(u + n))/(Γ(u))), when k � 1. Diaz and Pariguan [13] were motivated by the importance of (u) n,k , and they called it "the Pochhammer k− symbol," and they introduced the k− gamma function by which satisfies the functional equation: As special cases, Γ 1 (u) is the ordinary gamma function and the case k � 2 is of particular interest since Γ 2 (u) � ∞ 0 v u− 1 e − (v 2 /2) dv is the Gaussian integral. e k− gamma and ordinary gamma functions are related by the relation e k-analogue of the digamma function is given by [14]: and it satisfies the following relations for u and k > 0 and m � 0, 1, 2, . . . ,: In 2018, Nantomah et al. [15] presented the following integral representations: Ege and Yildirim [16] obtained an inequality involving ψ k ′ (u + k) and ψ k ″ (u + k) which can be written, by using (13), as In 2020, Yildirim [17] presented some monotonicity properties for ψ (m) k functions, and he deduced the following inequalities: For more information about Γ k and ψ k functions, we refer the reader to [18][19][20] and the related references therein.
Nantomah et al. [21] introduced the following two-parameter deformation of the gamma and digamma functions for u and k > 0 and p ∈ N: Nantomah et al. [22] introduced some complete monotonicity properties for Γ p,k , and they deduced the following inequalities: Journal of Mathematics Motivated by these results, we will present a k− analogue of the functions M α (u) and T δ (u) and study their monotonicity. As a consequence, we deduce some inequalities for Γ k and ψ (m) k functions.

Preliminary
Using relation (11), we have Also, using the following asymptotic formula of Γ function [4] ln with relation (11), we conclude for k > 0 that and for s ∈ N, In the following, we will use the following important result [23].
Now, we will prove the following auxiliary results.
us, we deduce that U β,k (x) is a completely monotonic function on (0, ∞). Conversely, if U β,k (x) is a completely monotonic function, then it is positive and we obtain Using asymptotic expansions (25) and (26), we have en,
Corollary 3. For a and b ≥ 0, the following inequality holds: with the best possible constants a � (1/3) and b � 0.
Remark 5. From Lemma 1, we conclude that the lower bound of (60) for m � 2 improves the lower bound of (16) for all x ≥ 0.83k.

Remark 6.
From Lemma 1, we conclude that the lower bound of (60) for m � 3 improves the lower bound of (19) for all x ≥ 0.76k.

Remark 8.
e upper bound of inequality (65) improves the upper bound of inequality (54) for all x > 0.
Proof. e left-hand side of inequality (67) is equivalent (− 1) m U (m) (1/4),k (x) > 0, and the right-hand side of inequality (67) is equivalent (− 1) m U (m) 0,k (x) < 0 in eorem 2. Data Availability e data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.