Certain Properties of the Modified Degenerate Gamma function

. In this paper, we prove some inequalities satisﬁed by the modiﬁed degenerate gamma function which was recently introduced. The tools employed include the Holder’s inequality, the mean value theorem, the Hermite-Hadamard’s inequality and the Young’s inequality. By some parameter varia-tions, the established results reduce to the corresponding results for the classical gamma function.


Introduction
In recent times, degenerate special functions and polynomials have been a subject of intense discussion. See for example [4], [5], [7], [8], [10] and the related references therein.
Derivatives of the modified degenerate gamma function are given as where r ∈ N 0 . In a recent work, He et al. [2] introduced the modified degenerate digamma function which is defined as and has the following representations among others.
where γ is the Euler-Mascheroni constant. It also satifies the following basic properties and similarly, it is clear that lim λ→0 ψ λ (x) = ψ(x) where ψ(x) is the classical digamma function. For further properties of the function ψ λ (x), one may refere to [2]. 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.

Results and Discussion
Theorem 2.1. For s ∈ (0, 1] and x > 0, the inequality holds. Proof. The case for s = 1 is obvious. So, let s ∈ (0, 1) and x > 0. Then by applying the Holder's inequality for integrals, we have and by using (6), we obtain By replacing s with 1 − s in (16), followed by substituting x by x + s, we obtain Now, combining (16) and (17) gives and by using (6), we obtain the desired results (15).
Remark 2.2. Inequality (18) can also be rearranged as which is the degenerate form of the Wendel's inequality (see (7) of [14]). Furthermore, by Squeezes theorem, (19) implies that which is the degenerate form of the Wendel's asymptotic relation (see (1) of [14]). The limit (20) also implies that holds.
Proof. The case for u = v is trivial. So, consider the function ln Γ λ (x) on the interval 0 < u < v. Then by the mean value theorem, there exist a k ∈ (u, v) such that and by exponentiation, we obtain the desired result (22). holds.
Proof. Let v = x + 1 and u = x + s in Theorem 2.3. holds.
Proof. Let 0 < u ≤ v and consider the function ψ λ (x) on the interval [u, v]. Since ψ λ (x) is concave, then by the classical Hermite-Hadamard inequality, we have which translates to and by exponentiation, we obtain the desired result (25).
(28) The upper bound of (28) coincides with the upper bound of inequality (1.2) in the work [3] which was obtained by a different procedure. However, the lower bound of (28) is better than the lower bound of inequality (1.2) in [3] since This is by virtue of the arithmetic-geometric mean inequality and the monotonicity property of ψ(x).
Proof. By the Holder's inequality for integrals, we have which concludes the proof.
Proof. By using (8) and the Holder's inequality, we have which concludes the proof.
Remark 2.17. If r 1 = r 2 = r, then (33) reduces to which implies that the function (8) is log-convex for any even order derivative. Moreover, if r = 0 in (34), we obtain which shows that the modified degenerate gamma function is log-convex.
hold for x > 0.
Proof. We adopt the technique of Mortici [12] to estimate the function and by exponentiation, we arrive at (41).

Concluding Remarks
In this work, we have proved several inequalities satisfied by the modified degenerate gamma function which was recently introduced. When λ → 0, then the established results reduce to the correponding 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 is no conflict of interests regarding the publication of this paper.