#### Abstract

In this paper, we presented two completely monotonic functions involving the generalized gamma function and its logarithmic derivative , and established some upper and lower bounds for in terms of .

#### 1. Introduction

The ordinary gamma function is given by the following equation [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]. The digamma function is the logarithmic derivative of the gamma function and is given by [4]:where is Euler–Mascheroni’s constant. A function defined on an interval is said to be completely monotonic if it possesses derivatives for all such that

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 functionis completely monotonic on if and only if , and they also proved the function is completely monotonic on if and only if As a consequence, they introduced the following sharp bounds for function in terms of function:with the best possible constants and . In order to refine inequality (5), Batir [9] modified the function and proved that the functionwhich is completely monotonic on if and only if , and the function is completely monotonic on if and only if As a consequence, he deduced the following refinement of the inequality (5):with the best possible constants and .

In many contexts, such as the combinatorics of creation, the annihilation operators, and the perturbative computation of Feynam integrals, the following symbol appears [10–12]:which is a generalization of the ordinary Pochhammer symbol , when . Diaz and Pariguan [13] were motivated by the importance of , and they called it “the Pochhammer symbol,” and they introduced the gamma function bywhich satisfies the functional equation:

As special cases, is the ordinary gamma function and the case is of particular interest since is the Gaussian integral. The gamma and ordinary gamma functions are related by the relation

The -analogue of the digamma function is given by [14]:and it satisfies the following relations for and :

In 2018, Nantomah et al. [15] presented the following integral representations:

Ege and Yildirim [16] obtained an inequality involving and which can be written, by using (13), as

In 2020, Yildirim [17] presented some monotonicity properties for functions, and he deduced the following inequalities:

For more information about and functions, we refer the reader to [18–20] and the related references therein.

Nantomah et al. [21] introduced the following two-parameter deformation of the gamma and digamma functions for and :

Nantomah et al. [22] introduced some complete monotonicity properties for , and they deduced the following inequalities:

Motivated by these results, we will present a analogue of the functions and and study their monotonicity. As a consequence, we deduce some inequalities for and functions.

#### 2. Preliminary

Using relation (11), we have

Also, using the following asymptotic formula of function [4]with relation (11), we conclude for thatand for

In the following, we will use the following important result [23].

Corollary 1. *Let be a real-valued function defined on ; with tends to zero as . Then, for , , if for all and , if for all .*

*Now, we will prove the following auxiliary results.*

Lemma 1. *For , we have*

*Proof. *Consider the function , and using (13), we getand using asymptotic expansion (27), we have . Then, Corollary 1 gives us that for all . Now, we let and use (13) to getwhereThen, for all with ; hence, we have for all . It follows that for all with . Then, for all . Finally, consider the function and use (13) to obtainwherefor . Then, with . Hence, for all .

Using relation (12) of , we conclude the following corollary which will be useful in the next part.

Corollary 2. *Let and Then, the following limits are valid:and*

#### 3. Two Completely Monotonic Functions Involving and Functions

Theorem 1. *The functionis completely monotonic on if and only if , and the function is completely monotonic on if and only if *

*Proof. *Using relation (15), we getwhereLet then we obtainConsequently, is completely monotonic on ; hence, is increasing on . Using asymptotic expansions (26) and (27), we have , and then, Hence, is decreasing on . Using asymptotic expansions (25) and (26), we have ; hence, . Therefore, is completely monotonic on . Conversely, if is completely monotonic, then we getFrom (25), we haveUsing asymptotic expansion (26), we get Then, from (40), we conclude that ; hence, Now, for we haveand therefore, is completely monotonic on . Thus, is decreasing on with ; then, Thus, is increasing on with ; hence, . Then, is completely monotonic on . Conversely, we assume that (with ) is completely monotonic on ; then, is negative on . But this contradicts that ; hence,

Theorem 2. *The functionis completely monotonic on if and only if Also, is completely monotonic on if and only if *

*Proof. *Using relation (15), we getwhereLet then we obtainwhereThen, is completely monotonic on . Thus, is an increasing function on ; using asymptotic expansions (26) and (27), we get that . Then, is a decreasing function with ; hence, . Thus, we deduce that is a completely monotonic function on . Conversely, if is a completely monotonic function, then it is positive and we obtainUsing asymptotic expansions (25) and (26), we haveThen,Now, for , thenand hence, is a completely monotonic function on for . Now, is a decreasing function on with , and then, Thus, is an increasing function on with ; hence, . Therefore, we deduce that is a completely monotonic function on for Conversely, if is a completely monotonic function, then andUsing functional relations (10) and (13), we get , where

##### 3.1. Some Inequalities for , and Functions

As a consequence of the completely monotonicity properties of the two functions and , we conclude the following results.

Corollary 3. *For , the following inequality holds:with the best possible constants and *

*Proof. *The left-hand side of inequality (54) is equivalent which leads to as mentioned in the proof of Theorem 1. Using the increasing property of on we get for , and then, is the best possible constant in (54). Also, Theorem 1 gives the right-hand side of inequality (54) for . If there exists such that the right-hand side of (54) is valid for then we would havewhich contradicts that Hence, is the best possible constant in (54).

*Remark 1. *If we put in inequality (54), then we get inequality (5).

Corollary 4. *For , the following inequality holds:with the best possible constants and .*

*Proof. *The left-hand side of inequality (56) is equivalent thatwhere . Using asymptotic expansions (26) and (27), we have andHence, we conclude from (57) that , and then, . Using the decreasing property of on we get that is the best possible constant in (56). Also, Theorem 1 gives the right-hand side of inequality (56) for If there exists such that the right-hand side of (56) is valid for then we would haveFrom (34) and (35), we have and which contradict with (59). Hence, the best possible constant in (56) is

*Remark 2. *The lower bound of (56) improves the lower bound of (21), when , for all

*Remark 3. *From Lemma 1, we conclude that the lower bound of (56) improves the lower bound of (17) for all

Corollary 5. *For , , and the following inequality holds:with the best possible constants and *

*Proof. *For the left-hand side of (60) is equivalent thatwhereUsing asymptotic expansion (27), we have andHence, we conclude from (61) that , and this leads to . Since is strictly completely monotonic on then the function is increasing on for and then, is the best possible constant in (60). Also, Theorem 1 gives the right-hand side of inequality (60) for If there exists such that the right-hand side of (60) is valid for then we would haveFrom (34) and (35), we have and also , and this contradicts with (64). Then, the best possible constant in (60) is

*Remark 4. *For the lower bound of (60) improves the lower bound of (22), when , for all .

*Remark 5. *From Lemma 1, we conclude that the lower bound of (60) for improves the lower bound of (16) for all

*Remark 6. *From Lemma 1, we conclude that the lower bound of (60) for improves the lower bound of (19) for all

Corollary 6. *The following inequality holds:*

*Proof. *The left-hand side of inequality (65) is equivalent , and the right-hand side of inequality (65) is equivalent in Theorem 2.

*Remark 7. *If we let in inequality (65), then we obtain inequality (7).

*Remark 8. *The upper bound of inequality (65) improves the upper bound of inequality (54) for all

Corollary 7. *The following inequality holds:*

*Proof. *The left-hand side of inequality (66) is equivalent , and the right-hand side of inequality (66) is equivalent in Theorem 2.

*Remark 9. *The upper bound of inequality (66) improves the upper bound of inequality (56) for all

Corollary 8. *The following inequality holds:*

*Proof. *The left-hand side of inequality (67) is equivalent , and the right-hand side of inequality (67) is equivalent in Theorem 2.

*Remark 10. *The upper bound of inequality (67) improves the upper bound of inequality (60) for all

#### Data Availability

The data used to support the findings of the study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.