Completely monotonic degree of a function involving the tri- and tetra-gamma functions

Let $\psi(x)$ be the di-gamma function, the logarithmic derivative of the classical Euler's gamma function $\Gamma(x)$. In the paper, the author shows that the completely monotonic degree of the function $[\psi'(x)]^2+\psi''(x)$ is $4$, surveys the history and motivation of the topic, supplies a proof for the claim that a function $f(x)$ is strongly completely monotonic if and only if the function $xf(x)$ is completely monotonic, conjectures the completely monotonic degree of a function involving $[\psi'(x)]^2+\psi''(x)$, presents the logarithmic concavity and monotonicity of an elementary function, and poses an open problem on convolution of logarithmically concave functions.


Introduction
A function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and 0 ≤ (−1) k−1 f (k−1) (x) < ∞ for x ∈ I and k ∈ N, where f (0) (x) means f (x) and N is the set of all positive integers. See [24, Chapter XIII], [48,Chapter 1], and [52,Chapter IV].
Let f (x) be a nonnegative function and have derivatives of all orders on (0, ∞). A number r ∈ R∪{±∞} is said to be the completely monotonic degree of f (x) with respect to x ∈ (0, ∞) if x r f (x) is a completely monotonic function on (0, ∞) but x r+ε f (x) is not for any positive number ε > 0. For convenience, we use the notation deg x cm [f (x)] to denote the completely monotonic degree r of f (x) with respect to x ∈ (0, ∞). For simplicity, in what follows, we sometimes just say that deg x cm [f (x)] is the degree of f (x). For more information on this notion, see [10,30,45] and related references therein.
The classical Euler's gamma function Γ(x) may be defined for x > 0 by The logarithmic derivative of Γ(x), denoted by ψ(x) = Γ ′ (x) Γ(x) , is called the psi or di-gamma function, the derivatives ψ ′ (x) and ψ ′′ (x) are respectively called the triand tetra-gamma functions. As a whole, hereafter, the derivatives ψ (i) (x) for i ≥ 0 are called polygamma functions.
The purpose of this paper is to compute the completely monotonic degree of the function with respect to x ∈ (0, ∞). We may state our main result as the following theorem.
In Section 3, we will prove our Theorem 1.1. In the last section, among other things, we will survey something to do with the function Ψ(x) and introduce the motivation of this paper.

Lemmas
In order to smoothly prove our Theorem 1.1, we need the following lemmas.  51]). Let f i (t) for i = 1, 2 be piecewise continuous in arbitrary finite intervals included in (0, ∞) and suppose that there exist some constants ∂f (x, t) ∂t dx.   [50]). and Remark 2.1. There have been some new refinements and generalizations of the famous Hermite-Hadamard inequality. See [4,5,46,53,54,56] and cited references therein.
Lemma 2.6. If f (x) is differentiable and logarithmically concave (or logarithmically convex respectively) on (−∞, ∞), then the product f (x)f (λ − x) for any fixed number λ ∈ R is increasing (or decreasing respectively) with respect to x ∈ −∞, λ 2 and decreasing (or increasing respectively) with respect to x ∈ λ 2 , ∞ . Proof. Taking the logarithm of f (x)f (λ − x) and differentiating give In virtue of the logarithmic concavity of f (x), it follows that the function is decreasing and For the case of f (x) being logarithmically convex, it may be proved similarly.
Remark 2.2. The techniques in the proof of Lemma 2.6 has been utilized in the papers [12,32,41,59] and closely related references therein.

Remarks
In this section we demonstrate the motivation of this paper by retrospecting the history and mentioning some known results related to our Theorem 1.1.   [9,28,35,38,39]. For more information about the history and background of this topic, please refer to the expository and survey articles [26,42] and plenty of references therein.
In [57,Theorem 1] and [58,Theorem 2], the functions are proved to be completely monotonic on (0, ∞). From this, we obtain max In [43,Theorem 1], the function was proved to be completely monotonic on (0, ∞) if and only if λ ≤ 0, and so is −h λ (x) if and only if λ ≥ 4; Consequently, the double inequality holds on (0, ∞) if and only if µ ≤ 0 and ν ≥ 4. The inequality (4.7) refines and sharpens the right-hand side inequality in (4.5). Motivated by the above results, we naturally pose the following problems: (1) Is the function f 4 (x) defined by (3.3) completely monotonic on (0, ∞)?
In other words, is the number 4 the completely monotonic degree of the function Ψ(x) with respect to x ∈ (0, ∞)? Our Theorem 1.1 of this paper affirmatively answers the above questions.  [18] implicitly. The divided difference form of the function Ψ(x) and related functions have been investigated in the papers [9,31,35,38,44] and closely related references therein. where a, b are positive numbers. We remark that some cases of the function g a,b (s) and its reciprocal have been investigated and applied in many articles such as [11,13,14,15,17,22,25,27,29,30,32,33,36,37,45,47,55] and closely-related references therein.
Remark 4.5. Recall from [49] that a function f is said to be strongly completely monotonic on (0, ∞) if it has derivatives of all orders and is nonnegative and decreasing on (0, ∞) for all n ≥ 0. In [21, p. 34, Proposition 1.1], it was claimed that a function f (x) is strongly completely monotonic if and only if the function xf (x) is completely monotonic. So any completely monotonic function on (0, ∞) of degree not less than 1 with respect to x must be strongly completely monotonic, and the degree with respect to x of any strongly completely monotonic function is not less than 1.
Since not finding a proof for [21, p. 34, Proposition 1.1], we here provide in detail a nontrivial and complete proof for it.