A class of completely monotonic functions involving the polygamma functions

Let Γ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Gamma (x)$\end{document} denote the classical Euler gamma function. We set ψn(x)=(−1)n−1ψ(n)(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\psi _{n}(x)=(-1)^{n-1}\psi ^{(n)}(x)$\end{document} (n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\in \mathbb{N}$\end{document}), where ψ(n)(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\psi ^{(n)}(x)$\end{document} denotes the nth derivative of the psi function ψ(x)=Γ′(x)/Γ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\psi (x)=\Gamma '(x)/\Gamma (x)$\end{document}. For λ, α, β∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta \in \mathbb{R}$\end{document} and m,n∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m,n\in \mathbb{N}$\end{document}, we establish necessary and sufficient conditions for the functions L(x;λ,α,β)=ψm+n(x)−λψm(x+α)ψn(x+β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L(x;\lambda ,\alpha ,\beta )=\psi _{m+n}(x)-\lambda \psi _{m}(x+ \alpha ) \psi _{n}(x+\beta ) $$\end{document} and −L(x;λ,α,β)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-L(x;\lambda ,\alpha ,\beta )$\end{document} to be completely monotonic on (−min(α,β,0),∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-\min (\alpha ,\beta ,0),\infty )$\end{document}. As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

A function f is said to be completely monotonic on an interval I if f has derivatives of all orders on I and (-1) n f (n) (x) ≥ 0, x ∈ I, n ≥ 0 (see [17]). A function f is said to be strictly completely monotonic if (-1) n f (n) (x) > 0. The Bernstein-Widder Theorem [17,Theorem 12b,p. 161] states that f is completely monotonic on (0, ∞) if and only if where α(t) is nondecreasing such that the integral converges for x > 0. Completely monotonic functions have attracted the attention of many researchers in various fields (see [8,[18][19][20][21][22][23][24]).
The aim of this paper is to solve this question and then apply it to obtain more inequalities involving ratios, differences of digamma and polygamma functions.
A detailed plan of this paper is as follows: In Sect. 2, we give detailed proof of our main results. In Sect. 3, some more inequalities for ratios of gamma functions are obtained with the aid of Theorem 3.1.
By the same argument, assertion (4) can be proved.
If β ≥ M and α ≥ β, then we write Together with U 1 (x) ≥ 0, it leads to (3.8). Similarly, we can prove that (3.8) is still valid for the case α ≥ M and β ≥ α.
Proof The sufficient conditions of the assertion is proved in the proof of Theorem 3.1.
Next we shall prove the necessary conditions. Suppose that α > 0. Since f 1 (x) and f 2 (x) are completely monotonic on (0, ∞), we have f 1 (x), f 2 (x) ≥ 0. On the other hand, it is easy to check that which yields contradictions. The proof is completed.

Remark 3 In [27, Theorem 2.2], Batir proved the inequality
for m ∈ N, n = 1, 2, . . . , m -1 and x > 0. By Theorem 3.1, inequality (3.11) can be refined partially. Taking the logarithm in (3.11) yields Corollary 2 For m and n ∈ N, we have the following inequalities Proof On the one hand, if m > n, a simple calculation shows that the right-hand side of (3.12) is equivalent to Similarly, the left-hand side of (3.12) is equivalent to By (3.15), (3.16) and Theorem 3.1, we see that (3.13) is proved.
On the other hand, if m < n, inequality (3.15) is reversed by a similar calculation. From the reversed inequality of (3.15), it follows that Taking into account the right-hand side of (3.12), the reversed inequality of (3.15), (3.17), and Theorem 3.1, we prove (3.14). Consequently, the proof of the two inequalities is complete.

Corollary 3
For β ∈ R and n ∈ N, let the function f 3 Proof A simple computation gives For β ≤ 0, from the right-hand side of (4.2), we get and therefore, in the view of Theorem 3.1, we have f 3 (x) < 0. By the same spirit, the lefthand side of (4.2) and Theorem 3.1 imply the case β ≥ 1 2 . This completes the proof.
In addition, it was proved in [36] that holds for x > -r if |t -s| < 1 and its reversed inequality is valid on (-r, ∞) if |t -s| > 1.
In the following, we will prove the monotonicity of the function z(x; λ, s, t) = (x; λ, s, t) × φ n (x) and therefore extend (4.5) or the right-hand side of (4.2).
Using Theorem 4.1, we have the following: Corollary 4 For s, t ∈ R, r = min{s, t} and n ∈ N, we have the inequality for x > -r if |t -s| < 1 and its reversed inequality is valid on (-r, ∞) if |t -s| > 1.
Proof Obviously, we only assume s = t. In view of Theorem 4.1, we only need to check According to [13,Corollary 1.4], the inequality holds for x > -r, so that this combined with (4.8) yields (4.7). Hence we complete the proof of this Theorem.  Proof Let g s,t (x) = e G s,t (x) , h s,t (x) = ln (x; 1, s, t) and f s,t (x) = g s,t (x)e x-h s,t (x)e h s,t (x) . Since g s,t (x) = g s,t (x)h s,t (x) and h s,t (x) = φ 1 (x), we obtain Using the asymptotic formula (see [4]) we get lim x→∞ h s,t (x) = ∞, and therefore by h s,t (x) > 0 and lim x→-r h s,t (x) = -∞, we conclude that 1 + h s,t (x) has a unique zero on (-r, ∞). Hence thanks to h s,t (x) > 0, Corollary 4 and (4.11), we have the following statements: (i) For |t -s| < 1, f s,t (x) is increasing on (X s,t , ∞) and decreasing on (-r, X s,t ), (ii) For |t -s| > 1, f s,t (x) is decreasing on (X s,t , ∞) and increasing on (-r, X s,t ).
On the one hand, we check that (4.13) Case 1. s = t. Now we derive the asymptotic formula of h s,t (x)e h s,t (x) . Taking the logarithm in (4.12), we get (4.14) Together with we can rewrite (4.14) as which implies (4.13). Case 2. s = t. Using (1.4) and the asymptotic formula (see [4]) we can easily prove (4.13).
On the other hand, we show that Note that the case t = s is obvious. Then using (4.15) and (4.19), we get which implies the exitance of constants C and X > 0 such that for all x > X. It follows that  Applying the monotonicity of f s,t (x) and (4.21), we complete the proof of this Theorem.

Discussion
Observing that Corollary 4 generalizes the right-hand side of (4.2), we conjecture that the left-hand side of (4.2) might be generalized to (n -1)! < x + 1 2 ; 1 n , s, t φ n (x) for x > -r if |t -s| < 1 and that its reversed inequality might be valid on (-r, ∞) if |t -s| > 1, where s, t ∈ R and r = min{s, t}. We turn to pay attention to the class of strongly completely monotonic functions, which are introduced in [40]. A function f : (0, ∞) → R is called strongly completely monotonic if it satisfies the more restrictive condition that (-1) n x n+1 f (n) (x) is nonnegative and decreasing on (0, ∞) for all n ∈ N. Note that [40,Theorem 1] gives a characterization of strongly completely monotonic functions.
It was shown in [20] that the function ψ 2 1 (x)ψ 2 (x) is strongly completely monotonic on (0, ∞). Inspired by this, we will determine necessary and sufficient conditions for λ such that the function (x; λ, s, t) is strongly completely monotonic on (-r, ∞) for all fixed s, t ∈ R and r = min{s, t} in the future work.