Abstract
In this paper, we prove some inequalities satisfied by the modified degenerate gamma function which was recently introduced. The tools employed include Holder’s inequality, mean value theorem, Hermite–Hadamard’s inequality, and Young’s inequality. By some parameter variations, the established results reduce to the corresponding results for the classical gamma function.
1. Introduction
In recent times, degenerate special functions and polynomials have been a subject of intense discussion. See, for example, [1–5] and the related references therein.
In 2017, Kim and Kim [6] introduced the degenerate gamma function aswhere and . This was motivated by the degenerate exponential function which is defined as [6]where . It is clear that and , where is the classical gamma function.
In 2018, Kim et al. [7] introduced the modified degenerate gamma function which is defined aswhere and . This definition is equivalent towhere and . Here, . The modified degenerate gamma function (3) satisfies the following properties [7]:
Derivatives of the modified degenerate gamma function are given aswhere .
In a recent work, He et al. [8] introduced the modified degenerate digamma function which is defined asand has the following representations among others:where is the Euler–Mascheroni constant. It also satisfies the following basic properties:and similarly, it is clear that , where is the classical digamma function. For further properties of the function , one may refer to [8].
In this paper, we continue to investigate the modified degenerate gamma function. Precisely, we prove some inequalities satisfied by this generalized function. The techniques we employed are analytical in nature.
2. Results and Discussion
Theorem 1. For and , the inequality holds:
Proof. The case for is obvious. So, let and . Then, by applying Holder’s inequality for integrals, we haveand by using (6), we obtainBy replacing with in (14), followed by substituting by , we obtainNow, combining (14) and (15), we obtainand by using (6), we obtain the desired results (12).
Remark 1. Inequality (16) can also be rearranged aswhich is the degenerate form of Wendel’s inequality (see (7) of [9]). Furthermore, by Squeezes theorem, (17) implies thatwhich is the degenerate form of Wendel’s asymptotic relation (see (1) of [9]). The limit (18) also implies that
Theorem 2. For , the inequality holds:
Proof. The case for is trivial. So, consider the function on the interval . Then, by the mean value theorem, there exist a such thatSince is increasing, thenwhich yieldsand by exponentiation, we obtain the desired result (20).
Corollary 1. For and , the inequality holds:
Proof. Let and in Theorem 2.
Corollary 2. For , the inequality holds:
Proof. Let and in Theorem 2.
Remark 2. Inequality (24) is the degenerate form of inequality (3.4) of [10].
Theorem 3. For , the inequality holds:
Proof. Let , and consider the function on the interval . Since is concave, then by the classical Hermite–Hadamard inequality, we havewhich translates toand by exponentiation, we obtain the desired result (26).
Corollary 3. For and , the inequality holds:
Proof. Let and in Theorem 3.
Corollary 4. For , the inequality holds:
Proof. Let and in Theorem 3.
Remark 3. Inequalities (26), (29), and (30) are, respectively, better than (20), (24), and (25)
Remark 4. By letting , inequality (29) reduces toThe upper bound of (31) coincides with the upper bound of inequality (1.2) in the work [11] which was obtained by a different procedure. However, the lower bound of (31) is better than the lower bound of inequality (1.2) in [11] sinceThis is by virtue of the arithmetic-geometric mean inequality and the monotonicity property of .
Theorem 4. For and , the inequality holds:
Proof. Let for and . Then,Also, inequality (17) implies thatFurthermore,which shows that is decreasing. Hence, which yields (33)
Remark 5. As a particular case, by letting and , we obtainThe upper bound of (37) agrees with the upper bound of (3) in [12]. Comparing the lower bounds of (37) and (3) in [12] reveals that the lower bound of (37) is stronger if and is weaker if .
Theorem 5. Let and . Then,holds for and .
Proof. By Holder’s inequality for integrals, we havewhich concludes the proof.
Remark 6. Applying Young’s inequality on the right-hand side of (38) reveals that
Theorem 6. Let , , , and . Then, inequality holds for and :
Proof. By using (8) and Holder’s inequality, we havewhich concludes the proof.
Remark 7. If , then (41) reduces towhich implies that function (8) is log-convex for any even order derivative. Moreover, if in (43), we obtainwhich shows that the modified degenerate gamma function is log-convex.
Remark 8. If , , , and , then (41) reduces to the Turan-type inequality:
Remark 9. If , , , and , then (35) reduces to the Turan-type inequality:where . This is the degenerate version of main result of [13].
Theorem 7. For , the inequality holds:
Proof. Let . Then,Thus, is increasing, and for , we have which completes the proof.
Lemma 1. The function is increasing for all .
Proof. By using (6) and (9) and the monotonicity property of , we haveand consequently, we obtainwhich completes the proof.
Theorem 8. The inequalities hold for :
Proof. By letting and replacing and with and in (44), we obtainAlso, since for , then . Now, let for . Then, by using (6), we haveNext, by using (5), (53), and (54), we obtainwhich gives (51). Next, let for . Then,It follows from Lemma 1 that is increasing if and decreasing if . For both cases, we have which gives inequality (52).
Remark 10. Inequality (52) can be obtained from inequality (51) by applying the arithmetic-geometric mean inequality.
Theorem 9. Let and . Then, inequality holds for :
Proof. We adopt the technique of Mortici [14] to estimate the functionThus,and by exponentiation, we arrive at (57).
3. Concluding Remarks
In this work, we have proved several inequalities satisfied by the modified degenerate gamma function which was recently introduced. When , the established results reduce to the corresponding results for the classical gamma function. It is our fervent hope that the present results will inspire further research on the modified degenerate gamma function.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.