Complete monotonicity properties of a function involving the polygamma function

In this paper, we study completete monotonicity properties of the function $f_{a,k}(x)=\psi^{(k)}(x+a) - \psi^{(k)}(x) - \frac{ak!}{x^{k+1}}$, where $a\in(0,1)$ and $k\in \mathbb{N}_0$. Specifically, we consider the cases for $k\in \{ 2n: n\in \mathbb{N}_0 \}$ and $k\in \{ 2n+1: n\in \mathbb{N}_0 \}$. Subsequently, we deduce some inequalities involving the polygamma functions.


Introduction and Preliminaries
The classical Gamma function, which is an extension of the factorial notation to noninteger values is usually defined as and satisfying the basic property Γ(x + 1) = xΓ(x), x > 0.
In [6], Qiu and Vuorinen established among other things that the function is strictly decreasing and convex on (0, ∞). Motivated by this result, Mortici [2] proved a more generalized and deeper result which states that, the function is strictly completely monotonic on (0, ∞). In this paper, the objective is to extend Mortici's results to the polygamma functions. Particularly, we study completete monotonicity properties of the function f a,k ( x k+1 , where a ∈ (0, 1) and k ∈ N 0 , by considering the cases for k ∈ {2n : n ∈ N 0 } and k ∈ {2n + 1 : n ∈ N 0 }. Unlike Mortici's work, the techniques of the present work are simple and do not rely on the Hausdorff-Bernstein-Widder theorem.

Main Results
We present our findings in this section by starting with the following lemma.
Recall that a function f : (0, ∞) → R is said to be completely monotonic on (0, ∞) if f has derivatives of all order and (−1) n f (n) (x) ≥ 0 for all x ∈ (0, ∞) and n ∈ N 0 . Let a ∈ (0, 1) and k ∈ {2n : n ∈ N 0 }. Then, by repeated differentiation and by using (2) and (4), we obtain as a result of Lemma 2.2. Alternatively, we could proceed as follows.
Notice that, since the function 1−e −t t is strictly decreasing on (0, ∞), then for a ∈ (0, 1), we have 1−e −at at > 1−e −t t . Remark 2.4. Since every completely monotonic function is convex and decreasing, it follows that f a,k (x) is strictly convex and strictly decreasing on (0, ∞).

Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.