Bounds and Completely Monotonicity of Some Functions Involving the Functions ψ′(l) and ψ″(l)

Abstract

Symmetrical patterns exist in the nature of inequalities, which play a basic role in theoretical and applied mathematics. In several studies, inequalities present accurate approximations of functions based on their symmetry properties. In the paper, we prove the completely monotonic (CM) property of some functions involving the function $\Delta \left(l\right)={\psi }^{″}\left(l\right)+{\left[{\psi }^{\prime }\left(l\right)\right]}^2$ and hence we deduce a new double inequality for it. Additionally, we study the CM degree of some functions involving the function ${\psi }^{\prime }\left(l\right)$. Our new bounds takes priority over some of the recently published results.

1. Introduction and Preliminaries

The Psi (or Digamma) function $\psi$ is defined by [1]

$$\psi \left(l\right)=\frac{{\Gamma }^{\prime }\left(l\right)}{\Gamma \left(l\right)},$$

(1)

where the ordinary Euler’s gamma function $\Gamma$ is defined

$$\Gamma \left(l\right)={\int }_0^{\infty }{\mathrm{e}}^{-\alpha }{\alpha }^{l-1}\mathrm{d}\alpha ,l>0.$$

(2)

The two functions ${\psi }^{\prime }\left(l\right)$ and ${\psi }^{″}\left(l\right)$ are called the tri- and tetra-gamma functions, respectively. The polygamma functions ${\psi }^{\left(r\right)}\left(l\right)$ have several applications in special functions, applied mathematics and physics [2]. For example, several series involving polygamma functions formed in Feynman’s calculations [3] and in some mesh-free numerical methods, the digamma function use as family of basis functions for solving some partial differential equations [4].

For a function K defined on $l\ge 0$, if the derivatives $K^{\left(r\right)}\left(l\right)$ exist for all $r\ge 0$ with

$${(-1)}^r{K}^{\left(r\right)}\left(l\right)\ge 0,l\ge 0;r\ge 0$$

then it is called CM [5] on the interval $l\ge 0$. Bernstein–Widder theorem states that [6], the function $K\left(l\right)$ is CM on $l\ge 0$ if, and only if, it is a Laplace transform of a bounded and non-decreasing measure.

Alzer deduced the inequality [7]

$$\frac{h\left(l\right)}{900l^4{(l+1)}^{10}}<\Delta \left(l\right),l>0$$

(3)

where

$$\begin{array}{ccc}\hfill h\left(l\right)& =& 450+3600l+13290l^2+29700l^3+44101l^4+45050l^5+31865l^6+15370l^7\hfill \\ & & +4840l^8+900l^9+75l^{10}.\hfill \end{array}$$

Guo et al. presented the CM property of the function [8]

$$\Delta \left(l\right)-\frac{h\left(l\right)}{900l^4{(l+1)}^{10}},l>0.$$

Zhao et al. deduced the CM property of the functions [9]

$$\Delta \left(l\right)-\frac{l^2+12}{12{(l+1)}^2{l}^4}\mathrm{and}-\Delta \left(l\right)+\frac{l+12}{12(l+1)l^4},l>0.$$

Additionally, they deduced the inequality

$$\frac{l^2+12}{12{(1+l)}^2{l}^4}<\Delta \left(l\right)<\frac{l+12}{12(1+l)l^4},l>0.$$

(4)

The lower bounds of the inequalities (3) and (4) overlap each other.

Qi deduced the CM property of the function [10]

$$\Delta \left(l\right)-\frac{l^2+\mu l+12}{12{(l+1)}^2{l}^4},l>0$$

if, and only if, $\mu \le 0$ and so its negative if, and only if, $\mu \ge 4$. Additionally, he proved

$$\frac{l^2+\omega l+12}{12{(l+1)}^2{l}^4}<\Delta \left(l\right)<\frac{l^2+\nu l+12}{12{(l+1)}^2{l}^4},l>0$$

(5)

if, and only if, $\omega \le 0$ and $\nu \ge 4$. In case of $\nu =4$ and $\omega =0$, inequality (5) refines the upper bound and recovers the lower bound and of inequality (4). In 2022, Anis et al. presented the double inequality [11]

$$\frac{l^2+3l+3}{3l^4{(2l+1)}^2}<\Delta \left(l\right)<\frac{625l^2+2275l+5043}{3l^4{(50l+41)}^2},l>0$$

(6)

which improves the upper bound of inequality (5) for $l>0$ and its lower bound for $l>\frac{\sqrt{849}+9}{32}$.

Qi and Agarwal [12] presented a survey about the function $\Delta \left(l\right)$ contains inequalities, generalizations, q-analogous, CM property, several applications and pose some open problems. For more results about the function $\Delta \left(l\right)$, we refer to [13,14,15,16,17,18,19,20,21,22] and the literature listed therein.

Let $K\left(l\right)$ be a CM function on $l>0$ and use the notation $K(\infty )={lim}_{l\to \infty }K\left(l\right)$. If $l^{\varrho }\left[K\left(l\right)-K(\infty )\right]$ is CM function on $l>0$ if, and only if, $\varrho \in [0,\eta ]$, then the number $\eta \in R$ is called the CM degree of $K\left(l\right)$ on $l>0$ and is denoted by ${\mathrm{deg}}_{\mathrm{cm}}^l\left[K\left(l\right)\right]=\eta$. This concept is more accurately in measuring the CM property [9,23,24].

Qi deduced that [25]

$${\mathrm{deg}}_{\mathrm{cm}}^l[\Delta \left(l\right)]=4$$

and Zhao et al. deduced that [9]

$${\mathrm{deg}}_{\mathrm{cm}}^l\left[\Delta \left(l\right)-\frac{\delta \left(l\right)}{1800{(l+2)}^{10}{(l+1)}^{10}{l}^2}\right]=1,$$

where $\delta \left(l\right)$ is a finite polynomial of degree 21 has positive coefficients. For more results about the concept of CM degree, we refer to [12,16,24,26,27,28], and the references therein.

In this paper, we prove the CM property of the functions

$$\Delta \left(l\right)-\frac{105l^4+210l^3+161l^2-42l-164}{1260l^8}\mathrm{and}-\Delta \left(l\right)+\frac{15l^2+30l+23}{180l^6},l>0$$

and deduce the inequality

$$\frac{105l^4+210l^3+161l^2-42l-164}{1260l^8}<\Delta \left(l\right)<\frac{15l^2+30l+23}{180l^6},l>0$$

which improve the lower bound of inequality (6) for $l\ge 1.2287$ and its upper bound for $l\ge 1.7254$. Additionally, we study the CM degree of two functions involving ${\psi }^{\prime }\left(l\right)$.

2. Studying of Two CM Functions Involving ${\mathit{\psi }}^{\prime }\left(\mathit{l}\right)$

In the following sequel, we will use the following Lemma [29]:

Lemma 1.

$$\frac{1}{l^{\kappa }}=\frac{1}{\Gamma \left(\kappa \right)}{\int }_0^{\infty }{\alpha }^{\kappa -1}{\mathrm{e}}^{-l\alpha }\mathrm{d}\alpha ,\kappa ,l>0.$$

(7)

Now, we will study two CM functions involving ${\psi }^{\prime }\left(l\right)$ and deduce some new bounds of it.

Lemma 2.

The function

$$\begin{array}{cc}\hfill P\left(l\right)& ={\psi }^{\prime }\left(l\right)-\frac{k\left(l\right)}{2520l^6{(l+1)}^8}\hfill \end{array}$$

(8)

is CM on $(0,\infty )$, where

$$\begin{array}{cc}\hfill k\left(l\right)& =2520l^{13}+21420l^{12}+81060l^{11}+179760l^{10}+258636l^9+252168l^8+168228l^7\hfill \\ & +74466l^6+16212l^5-6279l^4-8862l^3-4767l^2-1354l-164.\hfill \end{array}$$

Proof. 

Using the Formula (7) and

$${(-1)}^r{\psi }^{\left(r\right)}\left(l\right)={\int }_0^{\infty }\frac{{\alpha }^r{\mathrm{e}}^{-l\alpha }}{{\mathrm{e}}^{-\alpha }-1}\mathrm{d}\alpha ,r\in \mathbb{N},l>0,$$

(9)

we have

$$P\left(l\right)={\int }_0^{\infty }\frac{{\mathrm{e}}^{-\alpha }}{6350400\left({\mathrm{e}}^{\alpha }-1\right)}p\left(\alpha \right){\mathrm{e}}^{-l\alpha }\mathrm{d}\alpha ,l>0,$$

(10)

where

$$\begin{array}{ccc}\hfill p\left(\alpha \right)& & =82{\alpha }^7-1001{\alpha }^6-1701{\alpha }^5+16170{\alpha }^4+64680{\alpha }^3-264600\alpha +e^{2\alpha }(3444{\alpha }^5+4410{\alpha }^4\hfill \\ & & -67620{\alpha }^3-264600{\alpha }^2+2910600\alpha -6350400)+e^{\alpha }(-82{\alpha }^7+1001{\alpha }^6-1743{\alpha }^5\hfill \\ & & -20580{\alpha }^4+2940{\alpha }^3+264600{\alpha }^2+3704400\alpha +6350400)\hfill \\ & & =\left(3444{\alpha }^5+4410{\alpha }^4-67620{\alpha }^3-264600{\alpha }^2+2910600\alpha -6350400\right)\sum _{n=0}^{\infty }\frac{\left(2^n{\alpha }^n\right)}{n!}\hfill \\ & & +(-82{\alpha }^7+1001{\alpha }^6-1743{\alpha }^5-20580{\alpha }^4+2940{\alpha }^3+264600{\alpha }^2+3704400\alpha \hfill \\ & & +6350400)\sum _{n=0}^{\infty }\frac{{\alpha }^n}{n!}+82{\alpha }^7-1001{\alpha }^6-1701{\alpha }^5+16170{\alpha }^4+64680{\alpha }^3-264600\alpha \hfill \\ & & =\frac{11125{\alpha }^{10}}{6}+2472{\alpha }^9+1080{\alpha }^8+\sum _{n=11}^{\infty }\frac{{\lambda }_n{\alpha }^n}{n!}\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill {\lambda }_n& & =-82n^7+2723n^6+861n^5{2}^{n-3}-31108n^5+2205n^4{2}^{n-3}-861n^4{2}^{n-2}-861n^4{2}^n\hfill \\ & & +142205n^4+2583n^3{2}^{n-3}-2205n^3{2}^{n-2}-9555n^3{2}^n+861n^3{2}^{n+2}-292978n^3\hfill \\ & & +6615n^2{2}^{n-3}-2583n^2{2}^{n-2}+7161n^2{2}^n-861n^2{2}^{n+2}-735n^2{2}^{n+6}+535472n^2\hfill \\ & & -6615n{2}^{n-2}+188391n{2}^{n+3}+3348168n-\left(99225\right)2^{n+6}+6350400\hfill \\ & & =\frac{1}{8}(-656n^7+21784n^6+861n^5{2}^n-248864n^5-6405n^4{2}^n+1137640n^4-50715n^3{2}^n\hfill \\ & & -2343824n^3-345135n^2{2}^n+4283776n^2+6021897n{2}^{n+1}+26785344n-\left(99225\right)2^{n+9}\hfill \\ & & +50803200)\hfill \\ & & =\frac{1}{8}\left(4283776n^2+26785344n+50803200+{\lambda }_{1,n}+{\lambda }_{2,n}+{\lambda }_{3,n}\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill {\lambda }_{1,n}& & =82n^5{2}^n-656n^7=82n^5\left(2^n-8n^2\right)>0,n\ge 10\hfill \\ \hfill {\lambda }_{2,n}& & =21784n^6-248864n^5+1137640n^4-2343824n^3\hfill \\ & & =56n^3\left(389n^3-4444n^2+20315n-41854\right))>0,n\ge 6\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\lambda }_{3,n}& & =779n^5{2}^n-6405n^4{2}^n-50715n^3{2}^n-345135n^2{2}^n+6021897n{2}^{n+1}-\left(99225\right)2^{n+9}\hfill \\ & & =2^n\left(779n^5-6405n^4-50715n^3-345135n^2+12043794n-50803200\right)>0,n\ge 11.\hfill \end{array}$$

Then, ${\lambda }_n>0$ for $n\ge 11$ and, hence, $p\left(\alpha \right)>0$ for $\alpha >0$. Then, $P\left(l\right)$ is CM function for $l>0$. □

Corollary 1.

The following inequality holds

$$\frac{k\left(l\right)}{2520l^6{(l+1)}^8}<{\psi }^{\prime }\left(l\right),l>0,$$

(11)

where

$$\begin{array}{ccc}\hfill k\left(l\right)& & =2520l^{13}+21420l^{12}+81060l^{11}+179760l^{10}+258636l^9+252168l^8+168228l^7\hfill \\ & & +74466l^6+16212l^5-6279l^4-8862l^3-4767l^2-1354l-164.\hfill \end{array}$$

Proof. 

Using the decreasing property of $P\left(l\right)$, and the asymptotic expansion [29]

$${\psi }^{\prime }\left(l\right)\sim \frac{1}{l}+\frac{1}{2l^2}+\sum _{r=2}^{\infty }\frac{B_r}{l^{r+1}},l\to \infty$$

(12)

where the Bernoulli number $B_r$ is defined by [30]

$$\frac{z}{{\mathrm{e}}^z-1}=\sum _{r=0}^{\infty }\frac{B_r}{r!}{z}^r,\left|z\right|<2\pi$$

we get ${lim}_{l\to \infty }P\left(l\right)=0$ and hence $P\left(l\right)>0$ for $l>0$. □

Lemma 3.

The function

$$\begin{array}{ccc}\hfill Q\left(l\right)& & =\frac{j\left(l\right)}{360l^4{(l+1)}^6}-{\psi }^{\prime }\left(l\right)\hfill \end{array}$$

is CM for $l>0$, where

$$j\left(l\right)=360l^9+2340l^8+6540l^7+10260l^6+9888l^5+6030l^4+2440l^3+720l^2+168l+23.$$

Proof. 

Using the Formulas (7) and (9), we get

$$Q\left(l\right)={\int }_0^{\infty }\frac{{\mathrm{e}}^{-\alpha }}{43200\left({\mathrm{e}}^{\alpha }-1\right)}q\left(\alpha \right){\mathrm{e}}^{-l\alpha }\mathrm{d}\alpha ,l>0$$

(13)

where

$$\begin{array}{ccc}\hfill q\left(\alpha \right)& & =-23e^{\alpha }{\alpha }^5+23{\alpha }^5+80e^{\alpha }{\alpha }^4-80{\alpha }^4-20e^{\alpha }{\alpha }^3+460e^{2\alpha }{\alpha }^3-440{\alpha }^3-1800e^{\alpha }{\alpha }^2\hfill \\ & & +1800e^{2\alpha }{\alpha }^2-25200e^{\alpha }\alpha -19800e^{2\alpha }\alpha +1800\alpha -43200e^{\alpha }+43200e^{2\alpha }\hfill \\ & & =23{\alpha }^5-80{\alpha }^4-440{\alpha }^3+1800\alpha +20\sum _{n=0}^{\infty }\frac{2^n{\alpha }^n}{n!}\left(23{\alpha }^3+90{\alpha }^2-990\alpha +2160\right)\hfill \\ & & -\sum _{n=0}^{\infty }\frac{{\alpha }^n}{n!}\left(23{\alpha }^5-80{\alpha }^4+20{\alpha }^3+1800{\alpha }^2+25200\alpha +43200\right)\hfill \\ & & =\frac{126591853{\alpha }^{17}}{1482030950400}+\frac{1153427{\alpha }^{16}}{1981324800}+\frac{4982647{\alpha }^{15}}{1362160800}+\frac{23559{\alpha }^{14}}{1121120}+\frac{161929{\alpha }^{13}}{1482624}\hfill \\ & & +\frac{6019{\alpha }^{12}}{11880}+\frac{32629{\alpha }^{11}}{15840}+\frac{36059{\alpha }^{10}}{5040}+\frac{2573{\alpha }^9}{126}+\frac{314{\alpha }^8}{7}+\frac{1391{\alpha }^7}{21}+42{\alpha }^6+\sum _{n=18}^{\infty }\frac{{\tau }_n{\alpha }^n}{n!}\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill {\tau }_n& & =-23n^5+310n^4+115n^3{2}^{n-1}-1305n^3+555n^2{2}^{n-1}+290n^2\hfill \\ & & -10235n{2}^n-24472n+\left(675\right)2^{n+6}-43200={\tau }_{1,n}+{\tau }_{2,n}+{\tau }_{3,n}.\hfill \end{array}$$

Now

$${\tau }_{3,n}=290n^2-24472n+\left(675\right)2^{n+6}-43200>43490n^2-24472n-43200>0,n>1.$$

and

$${\tau }_{2,n}=35n^3{2}^{n-1}+555n^2{2}^{n-1}-10235n{2}^n=5n{2}^{n-1}\left(7n^2+111n-4094\right)>0,n\ge 18.$$

Additionally, using the inequality $n^2<2^n$, we get

$$\begin{array}{ccc}\hfill {\tau }_{1,n}& & =-23n^5+310n^4+80n^3{2}^{n-1}-1305n^3=-n^3\left(23n^2-310n-\left(40\right)2^n+1305\right)\hfill \\ & & >n^3\left(17n^2+310n-1305\right)>0,n\ge 4.\hfill \end{array}$$

Then, $q\left(v\right)>0$ for $v>0$, and hence the function $Q\left(l\right)$ is CM for $l>0$. □

Corollary 2.

The following inequality holds

$$\frac{j\left(l\right)}{360l^4{(l+1)}^6}>{\psi }^{\prime }\left(l\right),l>0.$$

where

$$j\left(l\right)=360l^9+2340l^8+6540l^7+10260l^6+9888l^5+6030l^4+2440l^3+720l^2+168l+23.$$

Proof. 

Using the decreasing property of $Q\left(l\right)$, and the asymptotic expansion (12), we get ${lim}_{l\to \infty }Q\left(l\right)=0$ and then $Q\left(l\right)>0$ for $l>0$. □

3. Two CM Functions Involving $\mathbf{\Delta }\left(\mathit{l}\right)$ and Its New Inequality

In this section, we will study two CM functions involving $\Delta \left(l\right)$ and hence we will deduce a new inequality for this function.

Theorem 1.

The functions

$$H_1\left(l\right)=\Delta \left(l\right)-\frac{105l^4+210l^3+161l^2-42l-164}{1260l^8}$$

(14)

and

$$H_2\left(l\right)=-\Delta \left(l\right)+\frac{15l^2+30l+23}{180l^6}$$

(15)

are CM on $(0,\infty )$.

Proof. 

Using recursion formula [29]

$$\psi (l+1)-\psi \left(l\right)=\frac{1}{l},l>0$$

(16)

we have

$$\begin{array}{ccc}\hfill H_1\left(l\right)-H_1(l+1)=& & \left\{{\psi }^{\prime }\left(l\right)-{\psi }^{\prime }(l+1)\right\}\left\{{\psi }^{\prime }(l+1)+{\psi }^{\prime }\left(l\right)\right\}\hfill \\ & & +\left\{{\psi }^{″}\left(l\right)-{\psi }^{″}(l+1)\right\}\hfill \\ & & -\frac{1}{1260l^8{(l+1)}^8}(420l^{11}+3360l^{10}+11676l^9+22848l^8+27108l^7\hfill \\ & & +19026l^6+3612l^5-7539l^4-8862l^3-4767l^2-1354l-164)\hfill \\ \hfill =& & \frac{2}{l^2}P\left(l\right).\hfill \end{array}$$

The CM property is closed under multiplication and, hence, the product of the functions $\frac{2}{l^2}$ and $P\left(l\right)$ is CM for $l>0$. Hence the difference $H_1\left(l\right)-H_1(l+1)$ is CM for $l>0$, and then the function $H_1\left(l\right)$ is also CM for $l>0$, see [10]. Now, using the recursion formula (16), we get

$$\begin{array}{ccc}\hfill H_2\left(l\right)-H_2(l+1)=& & \left\{{\psi }^{\prime }(l+1)-{\psi }^{\prime }\left(l\right)\right\}\left\{{\psi }^{\prime }\left(l\right)+{\psi }^{\prime }(l+1)\right\}\hfill \\ & & +\left\{{\psi }^{″}(l+1)-{\psi }^{″}\left(l\right)\right\}\hfill \\ & & +\frac{60l^7+360l^6+888l^5+1170l^4+1000l^3+540l^2+168l+23}{180l^6{(l+1)}^6}\hfill \\ \hfill =& & \frac{2}{l^2}Q\left(l\right).\hfill \end{array}$$

Hence, the functions $H_2\left(l\right)-H_2(l+1)$, and $H_2\left(l\right)$ are CM for $l>0$. □

Corollary 3.

The following inequality holds

$$\frac{105l^4+210l^3+161l^2-42l-164}{1260l^8}<\Delta \left(l\right)<\frac{15l^2+30l+23}{180l^6},l>0.$$

(17)

Proof. 

Using the Formula (12) and its derivative

$${\psi }^{″}\left(l\right)\sim -\frac{1}{l^2}-\frac{1}{l^3}-\sum _{r=2}^{\infty }\frac{(r+1)B_r}{l^{r+2}},l\to \infty$$

(18)

we obtain ${lim}_{l\to \infty }{H}_1\left(l\right)=0$ and then $H_1\left(l\right)>0$ for $l>0$, that is

$$\frac{105l^4+210l^3+161l^2-42l-164}{1260l^8}<\Delta \left(l\right),l>0.$$

Additionally, using the Formulas (12) and (18), we get ${lim}_{l\to \infty }{H}_2\left(l\right)=0$, and hence $H_2\left(l\right)>0$ for $l>0$, that is

$$\Delta \left(l\right)<\frac{15l^2+30l+23}{180l^6},l>0.$$

 □

Remark 1.

The upper bound of inequality (17) is better than the upper bound of inequality (6) for $x>\frac{41\left(1765+12\sqrt{37105}\right)}{96865}\approx 1.7254$. Additionally, the lower bound of inequality (17) is better than the lower bound of inequality (6) for $x\ge 1.2287$. That is, our new bounds in inequality (17) takes priority over the bounds of inequality (6) except near the origin point.

4. Studying the CM Degrees of Two Functions Involving ${\mathit{\psi }}^{\prime }\left(\mathit{l}\right)$

Theorem 2.

The CM degree of the function $P\left(l\right)$ is 1 on $(0,\infty )$.

Proof. 

Using the relation (10), we get

$$lP\left(l\right)={\int }_0^{\infty }\frac{{\mathrm{e}}^{-\alpha }}{635400{\left({\mathrm{e}}^{\alpha }-1\right)}^2}\theta \left(\alpha \right){\mathrm{e}}^{-l\alpha }\mathrm{d}\alpha ,l>0,$$

where

$$\begin{array}{ccc}\hfill \theta \left(\alpha \right)=& & 82{\alpha }^7-1575{\alpha }^6+4305{\alpha }^5+24675{\alpha }^4-194040{\alpha }^2-264600\alpha +264600\hfill \\ & & +420e^{3\alpha }\left(41{\alpha }^4+42{\alpha }^3-483{\alpha }^2-1260\alpha +6930\right)-2e^{\alpha }\left(82{\alpha }^7-1575{\alpha }^6\right.\hfill \\ & & \left.+4305{\alpha }^5+16065{\alpha }^4-8820{\alpha }^3-92610{\alpha }^2+1984500\right)+e^{2\alpha }\left(82{\alpha }^7-1575{\alpha }^6\right.\hfill \\ & & \left.+4305{\alpha }^5-9765{\alpha }^4-35280{\alpha }^3+211680{\alpha }^2-5556600\alpha +793800\right)\hfill \\ \hfill =& & \sum _{n=8}^{\infty }\frac{{\varsigma }_n{\alpha }^n}{n!}\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill {\varsigma }_n=& & 41n^7{2}^{n-6}-164n^7-609n^6{2}^{n-4}+6594n^6+19705n^5{2}^{n-5}-84560n^5\hfill \\ & & -144585n^4{2}^{n-5}+5740n^4{3}^{n-3}+442260n^4+674429n^3{2}^{n-6}-5600n^3{3}^{n-2}\hfill \\ & & -1066016n^3+1434783n^2{2}^{n-5}-598360n^2{3}^{n-3}+1361766n^2-1384460n{3}^{n-2}\hfill \\ & & -2829735n{2}^n-659880n+\left(99225\right)2^{n+3}+\left(107800\right)3^{n+3}-3969000\hfill \\ \hfill =& & -164n^7+6594n^6-84560n^5+442260n^4-1066016n^3+1361766n^2-659880n\hfill \\ & & -3969000+3^n\left(\frac{5740n^4}{27}-\frac{5600n^3}{9}-\frac{598360n^2}{27}-\frac{1384460n}{9}+2910600\right)\hfill \\ & & +2^n\left(\frac{41n^7}{64}-\frac{609n^6}{16}+\frac{19705n^5}{32}-\frac{144585n^4}{32}+\frac{674429n^3}{64}\right.\hfill \\ & & \left.+\frac{1434783n^2}{32}-2829735n+793800\right)\hfill \end{array}$$

$$\begin{array}{ccc}\hfill >& & -164n^7+6594n^6-84560n^5+442260n^4-1066016n^3+1361766n^2-659880n\hfill \\ & & -3969000+\left(\frac{5740n^4}{27}-\frac{5600n^3}{9}-\frac{598360n^2}{27}\right.\hfill \\ & & \left.-\frac{1384460n}{9}+2910600\right)n^2+\left(\frac{41n^7}{64}-\frac{609n^6}{16}+\frac{19705n^5}{32}-\frac{144585n^4}{32}\right.\hfill \\ & & \left.+\frac{674429n^3}{64}+\frac{1434783n^2}{32}-2829735n+793800\right)n^2\hfill \\ \hfill =& & \frac{41n^9}{64}-\frac{609n^8}{16}+\frac{14457n^7}{32}+\frac{1977101n^6}{864}-\frac{42995099n^5}{576}+\frac{401704261n^4}{864}\hfill \\ & & -\frac{36446219n^3}{9}+5066166n^2-659880n-3969000\hfill \\ \hfill =& & {\varsigma }_{1,n}+{\varsigma }_{2,n}+{\varsigma }_{3,n}.\hfill \end{array}$$

Now, using the inequalities

$${\varsigma }_{1,n}=\frac{41n^9}{64}-\frac{609n^8}{16}+\frac{14457n^7}{32}>0,n\ge 44$$

$$\begin{array}{ccc}\hfill {\varsigma }_{2,n}& & =\frac{1977101x^6}{864}-\frac{42995099x^5}{576}+\frac{401704261x^4}{864}-\frac{36446219x^3}{9}\hfill \\ & & =\frac{n^3\left(3954202n^3-128985297n^2+803408522n-6997674048\right)}{1728}>0,n\ge 28\hfill \end{array}$$

and

$${\varsigma }_{3,n}=5066166n^2-659880n-3969000>0,n\ge 1$$

we obtain that ${\varsigma }_n>0$ for $n\ge 44$. Furthermore, direct calculations by Mathematica software, ${\sum }_{n=8}^{44}{\varsigma }_n\frac{{\alpha }^n}{n!}$ has positive coefficients. Hence, ${\varsigma }_n>0$ for $n\ge 8$, and $\theta \left(\alpha \right)>0$ for $\alpha >0$. Then, $lP\left(l\right)$ is CM function for $l>0$ and we get

$${\mathrm{deg}}_{\mathrm{cm}}^x\left[P\left(l\right)\right]\ge 1.$$

(19)

If we suppose that $l^{\epsilon }P\left(l\right)$ is CM for $l>0$, then the function $l^{\epsilon }P\left(l\right)$ is decreasing, that is

$$\epsilon \le -\frac{l{P}^{\prime }\left(l\right)}{P\left(l\right)}=-\frac{x\left[{\xi }^{\prime }\left(l\right)-{\psi }^{″}\left(l\right)\right]}{\xi \left(l\right)-{\psi }^{\prime }\left(l\right)},$$

where

$$\xi \left(l\right)=\frac{s\left(l\right)}{2520l^6{(l+1)}^8}$$

with

$$\begin{array}{ccc}\hfill s\left(l\right)& & =2520l^{13}+21420l^{12}+81060l^{11}+179760l^{10}+258636l^9+252168l^8+168228l^7\hfill \\ & & +74466l^6+16212l^5-6279l^4-8862l^3-4767l^2-1354l-164.\hfill \end{array}$$

Using the relation [31]

$${\psi }^{\left(m\right)}\left(l\right)={(-1)}^{m+1}m!\sum _{j=0}^{\infty }\frac{1}{{(j+l)}^{r+1}},l>0;m=1,2,3,...$$

we get

$$\underset{l\to \infty }{lim}\left[-\frac{1}{l}+l{\psi }^{\prime }\left(l\right)\right]=1and\underset{l\to \infty }{lim}\left[\frac{2}{l}+l^2{\psi }^{″}\left(l\right)\right]=-1.$$

Additionally,

$$\underset{l\to \infty }{lim}\left[-\frac{1}{l}+l\xi \left(l\right)\right]=-1and\underset{l\to \infty }{lim}\left[\frac{2}{l}+l^2{\xi }^{\prime }\left(l\right)\right]=1.$$

Then,

$$\epsilon \le -\frac{\left[l^2{\psi }^{″}\left(l\right)+\frac{2}{l}\right]-\left[l^2{\xi }^{\prime }\left(l\right)+\frac{2}{l}\right]}{\left[l{\psi }^{\prime }\left(l\right)-\frac{1}{l}\right]-\left[l\xi \left(l\right)-\frac{1}{l}\right]}\to 1asl\to \infty$$

and hence we get

$${\mathrm{deg}}_{\mathrm{cm}}^l\left[P\left(l\right)\right]\le 1.$$

(20)

Combining inequalities (19) and (20) completes the proof. □

Theorem 3.

The CM degree of $Q\left(l\right)$ on $(0,\infty )$ satisfies

$$2\le {\mathrm{deg}}_{\mathrm{cm}}^l\left[Q\left(l\right)\right]<3.$$

Proof. 

Using the integral representation (10), we have

$$l^{2}Q\left(l\right)={\int }_0^{\infty }\frac{{\mathrm{e}}^{-\alpha }}{43200{\left({\mathrm{e}}^{\alpha }-1\right)}^2}\sigma \left(\alpha \right){\mathrm{e}}^{-l\alpha }\mathrm{d}\alpha ,l>0,$$

where

$$\begin{array}{ccc}\hfill \sigma \left(\alpha \right)=& & -23{\alpha }^5+310{\alpha }^4-660{\alpha }^3-1680{\alpha }^2+2760e^{\alpha }\alpha +840\alpha +3600e^{\alpha }+3600\hfill \\ \hfill =& & (2760\alpha +3600)\sum _{n=5}^{\infty }\frac{{\alpha }^n}{n!}+92{\alpha }^5+920{\alpha }^4+1320{\alpha }^3+2880{\alpha }^2+7200\alpha +7200\hfill \\ \hfill >& & 0.\hfill \end{array}$$

Then,

$${\mathrm{deg}}_{\mathrm{cm}}^l\left[Q\left(l\right)\right]\ge 2.$$

(21)

Additionally,

$$l^{3}Q\left(l\right)={\int }_0^{\infty }\frac{e^{-\alpha }}{43200}\chi \left(\alpha \right)e^{-l\alpha }dv,l>0$$

where

$$\begin{array}{ccc}\hfill \chi \left(\alpha \right)& =& -23{\alpha }^5-425{\alpha }^4+1900{\alpha }^3-300{\alpha }^2-4200\alpha +2760e^{\alpha }-2760.\hfill \end{array}$$

Using $\chi \left(1\right)=1740.46$ and $\chi \left(0.2\right)=-226.401$, we have

$${\mathrm{deg}}_{\mathrm{cm}}^l\left[Q\left(l\right)\right]<3.$$

(22)

Combining inequalities (21) and (22) completes the proof. □

5. Conclusions

The main conclusions of this paper are stated in Theorems 1–3. For the function $\Delta \left(l\right)={\psi }^{″}\left(l\right)+{\left[{\psi }^{\prime }\left(l\right)\right]}^2$, we investigate the complete monotonicity of the functions

$$H_1\left(l\right)=\Delta \left(l\right)-\frac{105l^4+210l^3+161l^2-42l-164}{1260l^8}\mathrm{and}{H}_2\left(l\right)=-\Delta \left(l\right)+\frac{15l^2+30l+23}{180l^6},l>0.$$

Hence, we present a double inequality for $\Delta \left(l\right)$. Additionally, for the two functions

$$\begin{array}{ccc}\hfill P\left(l\right)& =& {\psi }^{\prime }\left(l\right)-\frac{1}{2520l^6{(l+1)}^8}(2520l^{13}+21420l^{12}+81060l^{11}+179760l^{10}+258636l^9+252168l^8\hfill \\ & & +168228l^7+74466l^6+16212l^5-6279l^4-8862l^3-4767l^2-1354l-164),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill Q\left(l\right)& =& -{\psi }^{\prime }\left(l\right)+\frac{1}{360l^4{(l+1)}^6}(360l^9+2340l^8+6540l^7+10260l^6+9888l^5+6030l^4+2440l^3\hfill \\ & & +720l^2+168l+23)\hfill \end{array}$$

we prove that ${\mathrm{deg}}_{\mathrm{cm}}^l\left[P\left(l\right)\right]=1$ and $2\le {\mathrm{deg}}_{\mathrm{cm}}^l\left[Q\left(l\right)\right]<3$ and then we deduce double inequality of the function ${\psi }^{\prime }\left(l\right)$.