CERTAIN INEQUALITIES FOR THE CONFORMABLE GAMMA AND POLYGAMMA FUNCTIONS

In a recent paper, Sarikaya et al. introduced a new analogue of the classical Gamma function which they called, the conformable Gamma function. Motivated by their results, this paper seeks to establish some inequalities for the conformable Gamma and Polygamma functions. Among other analytical tools, the procedure relies on the generalized forms of some classical inequalities.


Introduction and Preliminaries
The classical Euler's Gamma function, which is an extension of the factorial notation to noninteger values may be defined for x > 0 by any of the following equivalent forms.
Euler gave another definition of the Gamma function which is called the panalogue or p-Gamma function. This is defined as where lim p→∞ Γ p (x) = Γ(x). Since then, several analogues (or generalizations) of the Gamma function have been introduced and investigated. See for instance [5], [3], [4], [6], [7], [9] and [8] for the various analogues. Then in a recent paper [10], the authors introduce another version of the Gamma function called the conformable Gamma function defined as follows.
where Γ k (x) is the k-analogue of the Gamma function.
This paper seeks to establish some inequalities for the conformable Gamma and Polygamma functions. We begin by recalling the following lemmas which shall be required in order to prove our results.
Remark 2.6. Lemma 2.5 is well-known in the literature, and the proofs Lemmas 2.1, 2.2, 2.3 and 2.4 can be found in pages 790-791 of [2].

Inequalities for the Conformable Gamma Function
This section is devoted to some inequalities involving the conformable Gamma function, Γ α,k (x).
Let ψ α,k (x) be the conformable Digamma function defined as the logarithmic derivative of Γ α,k (x). By direct computations using (18), we obtain where m ∈ N 0 and ψ Proof. These follow directly from (20).
is holds for x i > 0.
Proof. We make use of the Hermite-Hadamard's inequality which states that: if Let f (x) = −ψ α,k (x), a = x + u and b = x + 1. Then f (x) is convex and by (27) we obtain Then by taking exponents, we obtain the result (26).

Proof. By (8) we obtain
which completes the proof. Note: The first inequality in this proof follows from the fact that n i=1 a u i ≤ ( n i=1 a i ) u for a i ≥ 0, u ≥ 1 while the second inequality is as a result of the generalized Minkowski's inequality (14).

Inequalities for the Conformable Polygamma Function
In this section, we establish some inequalities for the conformable Polygamma function ψ (m) α,k (x).
holds for x i > 0.

Conclusion
Recently, Sarikaya et al. [10] introduced a new analogue of the Gamma function which they called the conformable Gamma function. Motivated by their results, we have established some inequalities involving the conformable Gamma and Polygamma functions. Apart from providing generalizations of some known results in the literature, the established results could trigger new research directions in the area of study.