# Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic

## Abstract

We prove that the function F α,β (x) = xαΓβ(x)/Γ(βx) is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) {(α, β) : β > 0, β ≥ 2α + 1, βα + 1}\{(α, β) : α = 0, β = 1} and that [F α,β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) {(α, β ) : β > 0, β ≤ 2α + 1, βα + 1}\{(α, β ) : α = 0, β = 1}.

2010 Mathematics Subject Classification: 33B15; 26A48.

## 1 Introduction

For real and positive values of x the Euler gamma function Γ and its logarithmic derivative ψ, the so-called digamma functions are defined by

$\Gamma \left(x\right)={\int }_{0}^{\infty }{t}^{x-1}{e}^{-t}dt,$
(1.1)
$\psi \left(x\right)=\frac{{\Gamma }^{\prime }\left(x\right)}{\Gamma \left(x\right)}=-\gamma +{\int }_{0}^{\infty }\frac{{e}^{-t}-{e}^{-xt}}{1-{e}^{-t}}dt,$
(1.2)

where γ = 0.5772 ··· is the Euler's constant.

For extension of these functions to complex variable and for basic properties see . Over the last half century, many authors have established inequalities and monotonicity for these functions .

We know that a real-valued function f : I is said to be completely monotonic on I if f has derivatives of all orders on I and

${\left(-1\right)}^{n}{f}^{\left(n\right)}\left(x\right)\ge 0$
(1.3)

for all x I and n ≥ 0. Moreover, f is said to be strictly completely monotonic if inequalities (1.3) are strict.

We also know that a positive real-valued function f : I → (0, ∞) is said to be logarithmically completely monotonic on I if f has derivatives of all orders on I and its logarithm log f satisfies

${\left(-1\right)}^{k}{\left[log\phantom{\rule{2.77695pt}{0ex}}f\left(x\right)\right]}^{\left(k\right)}\ge 0$
(1.4)

for all x I and k . Moreover, f is said to be strictly logarithmically completely monotonic if inequalities (1.4) are strict.

Recently, the completely monotonic or logarithmically completely monotonic functions have been the subject of intensive research. In particular, many complete monotonicity and logarithmically complete monotonicity properties related to the gamma function, psi function, and polygamma function can be found in the literature [17, 18, 2337]. In 1997, Merkle  proved that $F\left(x\right)=\frac{\Gamma \left(2x\right)}{{\Gamma }^{2}\left(x\right)}$ is strictly log-concave on (0, ∞). Later, Chen  showed that ${\left[F\left(x\right)\right]}^{-1}=\frac{{\Gamma }^{2}\left(x\right)}{\Gamma \left(2x\right)}$ is strictly logarithmically completely monotonic on (0, ∞). In , Li and Chen proved that ${F}_{\beta }\left(x\right)=\frac{{\Gamma }^{\beta }\left(x\right)}{\Gamma \left(\beta x\right)}$ is strictly logarithmically completely monotonic on (0, ∞) for β > 1, and that [F β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞) for 0 < β < 1. The purpose of this article is to generalize Li and Chen's result. Our main result is as follows.

Theorem 1.1 Let α , β > 0 and F α,β (x) = xα Γβ(x)/Γ(βx), then

1. (1)

F α,β (x) is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) {(α, β) : β > 0, β ≥ 2α + 1, βα + 1}\{(α, β) : α = 0, β = 1};

2. (2)

[F α,β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞) if and only if (α, β) {(α, β) : β > 0, β ≤ 2α + 1, βα + 1}\{(α, β) : α = 0, β = 1}.

## 2 Lemma

In order to prove our Theorem 1.1, we need a lemma which we present in this section.

Lemma 2.1 Let α , β (0, 1) (1, ∞) and

$h\left(t\right)=-\alpha {e}^{-\left(\beta +1\right)t}+\left(\alpha +1\right){e}^{-\beta t}+\left(\alpha -\beta \right){e}^{-t}+\beta -\alpha -1.$

Then the following statements are true:

1. (1)

If βα + 1 and β ≤ 2α + 1, then h(t) < 0 for t (0, ∞);

2. (2)

If α + 1 < β < 2α + 1, then there exists λ 1 (0, ∞) such that h(t) < 0 for t (0, λ 1) and h(t) > 0 for t (λ 1, ∞);

3. (3)

If βα + 1 and β ≥ 2α + 1, then h(t) > 0 for t (0, ∞);

4. (4)

If 2α + 1 < β < α + 1, then there exists λ 2 (0, ∞) such that h(t) > 0 for t (0, λ 2) and h(t) < 0 for t (λ 2, ∞).

Proof Let h1(t) = e(β + 1)th'(t) and ${h}_{2}\left(t\right)={e}^{-t}{h}_{1}^{\prime }\left(t\right)$. Then simple computations lead to

$h\left(0\right)=0,$
(2.1)
$\begin{array}{lll}\hfill {h}^{\prime }\left(t\right)& =\alpha \left(\beta +1\right){e}^{-\left(\beta +1\right)t}-\beta \left(\alpha +1\right){e}^{-\beta t}-\left(\alpha -\beta \right){e}^{-t},\phantom{\rule{2em}{0ex}}& \hfill \\ \hfill {h}_{1}\left(0\right)& ={h}^{\prime }\left(0\right)=0,\phantom{\rule{2em}{0ex}}& \hfill \\ \hfill \end{array}$
(2.2)
$\begin{array}{lll}\hfill {h}_{1}\left(t\right)& =\alpha \left(\beta +1\right)-\beta \left(\alpha +1\right){e}^{t}-\left(\alpha -\beta \right){e}^{\beta t},\phantom{\rule{2em}{0ex}}& \hfill \\ \hfill {{h}^{\prime }}_{1}\left(t\right)& =-\beta \left(\alpha +1\right){e}^{t}-\beta \left(\alpha -\beta \right){e}^{\beta t},\phantom{\rule{2em}{0ex}}& \hfill \\ \hfill \end{array}$
(2.3)
${h}_{2}\left(0\right)={h}_{1}^{\prime }\left(0\right)=\beta \left(\beta -2\alpha -1\right),$
(2.4)
${h}_{2}\left(t\right)=-\beta \left(\alpha +1\right)-\beta \left(\alpha -\beta \right){e}^{\left(\beta -1\right)t}$
(2.5)

and

${h}_{2}^{\prime }\left(t\right)=\beta \left(\beta -1\right)\left(\beta -\alpha \right){e}^{\left(\beta -1\right)t}.$
(2.6)
1. (1)

If βα + 1 and β ≤ 2α + 1, then we divide the proof into four cases.

Case 1 If 0 < β < 1 and α < β ≤ 2α + 1, then from (2.4) and (2.6) we clearly see that

${h}_{2}\left(0\right)\le 0,$
(2.7)
${h}_{2}^{\prime }\left(t\right)<0.$
(2.8)

Therefore, h(t) < 0 for t (0, ∞), which follows from (2.7) and (2.8) together with (2.1) and (2.2).

Case 2 If 0 < β < 1 and βα, then (2.5) and (2.6) lead to

$\underset{t\to +\infty }{lim}{h}_{2}\left(t\right)=-\beta \left(\alpha +1\right)<0,$
(2.9)
${h}_{2}^{\prime }\left(t\right)\ge 0.$
(2.10)

Therefore, h(t) < 0 for t (0, ∞), which follows from (2.9) and (2.10) together with (2.1) and (2.2).

Case 3 If 1 < βα, then (2.4) and (2.6) lead to

${h}_{2}\left(0\right)<0,$
(2.11)
${h}_{2}^{\prime }\left(t\right)\le 0.$
(2.12)

From equations (2.1) and (2.2) together with inequalities (2.11) and (2.12), we clearly see that h(t) < 0 for t (0, ∞).

Case 4 If β > 1 and α < βα + 1, then we clearly see that

$\underset{t\to +\infty }{lim}h\left(t\right)=\beta -\alpha -1\le 0.$
(2.13)

From (2.3)-(2.6), we know that

$\underset{t\to +\infty }{lim}{h}_{1}\left(t\right)=+\infty ,$
(2.14)
${h}_{2}\left(0\right)<0,$
(2.15)
$\underset{t\to +\infty }{lim}{h}_{2}\left(t\right)=+\infty ,$
(2.16)
${h}_{2}^{\prime }\left(t\right)>0.$
(2.17)

From (2.15)-(2.17), we clearly see that there exists t1> 0 such that h2(t) < 0 for t (0, t1) and h2(t) > 0 for t (t1, ∞). Hence, h1(t) is strictly decreasing in [0, t1] and strictly increasing in [t1, ∞).

From (2.2) and (2.14) together with the monotonicity of h1(t), we know that there exists t2> 0 such that h1(t) < 0 for t (0, t2) and h1(t) > 0 for t (t2, ∞). Hence, h(t) is strictly decreasing in [0, t2] and strictly increasing in [t2, ∞).

Therefore, h(t) < 0 for t (0, ∞) follows from (2.1) and (2.13) together with the monotonicity of h(t).

(2) If α + 1 < β < 2α + 1, then we clearly see that

$\underset{t\to +\infty }{lim}h\left(t\right)=\beta -\alpha -1>0$
(2.18)

and (2.14)-(2.17) hold again. From the proof of Case 4 in Lemma 2.1(1), we know that there exists λ > 0 such that h(t) is strictly decreasing in [0, λ] and strictly increasing in [λ, ∞).

Therefore, Lemma 2.1(2) follows from (2.1) and (2.18) together with the monotonicity of h(t).

(3) If βα + 1 and β ≥ 2α + 1, then we divide the proof into three cases.

Case I If β > 1 and β ≥ 2α + 1, then

$\beta >\alpha$
(2.19)

and it follows from (2.4) that

${h}_{2}\left(0\right)\ge 0.$
(2.20)

Equation (2.6) and inequality (2.19) lead to

${h}_{2}^{\prime }\left(t\right)>0.$
(2.21)

Therefore, h(t) > 0 for t (0, ∞) follows from (2.1), (2.2), (2.20), and (2.21).

Case II If 0 < β < 1 and α ≤ -1, then from (2.5) and (2.6) we clearly see that

$\underset{t\to +\infty }{lim}{h}_{2}\left(t\right)=-\beta \left(\alpha +1\right)\ge 0,$
(2.22)
${h}_{2}^{\prime }\left(t\right)<0.$
(2.23)

Inequalities (2.22) and (2.23) imply that

${h}_{2}\left(t\right)>0$
(2.24)

for t (0, ∞).

Therefore, h(t) > 0 for t (0, ∞) follows from (2.1) and (2.2) together with (2.24).

Case III If 0 < α + 1 ≤ β < 1, then we clearly see that

$\underset{t\to +\infty }{lim}h\left(t\right)=\beta -\alpha -1\ge 0.$
(2.25)

It follows from (2.3)-(2.6) that

$\underset{t\to +\infty }{lim}{h}_{1}\left(t\right)=-\infty ,$
(2.26)
${h}_{2}\left(0\right)=\beta \left(\beta -2\alpha -1\right)>0,$
(2.27)
$\underset{t\to +\infty }{lim}{h}_{2}\left(t\right)=-\beta \left(\alpha +1\right)<0,$
(2.28)
${h}_{2}^{\prime }\left(t\right)<0.$
(2.29)

Inequalities (2.27)-(2.29) imply that there exists t3> 0 such that h2(t) > 0 for t (0, t3) and h2(t) < 0 for t (t3, ∞). Hence, h1(t) is strictly increasing in [0, t3] and strictly decreasing in [t3, ∞).

It follows from (2.2) and (2.26) together with the monotonicity of h1(t) that there exists t4> 0 such that h1(t) > 0 for t (0, t4) and h1(t) < 0 for t (t4, ∞). Hence, h(t) is strictly increasing in [0, t4] and strictly decreasing in [t4, ∞).

Therefore, h(t) > 0 for t (0, ∞) follows from (2.1) and (2.25) together with the monotonicity of h(t).

(4) If 2α + 1 < β < α + 1, then we clearly see that

$\underset{t\to +\infty }{lim}h\left(t\right)=\beta -\alpha -1<0$
(2.30)

and (2.26)-(2.29) hold again.

From the proof of Case III in Lemma 2.1(3) we know that there exists μ > 0 such that h(t) is strictly increasing in [0, μ] and strictly decreasing in [μ, ∞).

Therefore, Lemma 2.1(4) follows from (2.1) and (2.30) together with the monotonicity of h(t).

## 3 Proof of Theorem 1.1

Proof of Theorem 1.1 Let E1 = {(α, β) : 0 < β < 1, βα + 1}, E2 = {(α, β) : β > 1, β ≥ 2α +1}, E3 = {(α, β) : α < 0, β = 1}, E4 = {(α, β) : α = 0, β = 1}, E5 = {(α, β) : α + 1 < β < 2α + 1}, E6 = {(α, β) : β > 0, 2α + 1 < β < α + 1}, E7 = {(α, β) : 0 < β < 1, β ≤ 2α + 1}, E8 = {(α, β) : β > 1, βα + 1} and E9 = {(α, β) : α > 0, β = 1}. Then

$\begin{array}{c}\left\{\left(\alpha ,\beta \right):\alpha \in ℝ,\beta >0\right\}={\cup }_{i=1}^{9}{E}_{i},\\ \left\{\left(\alpha ,\beta \right):\beta >0,\beta \ge 2\alpha +1,\beta \ge \alpha +1\right\}\\left\{\left(\alpha ,\beta \right):\alpha =0,\beta =1\right\}={E}_{1}\cup {E}_{2}\cup {E}_{3},\\ \left\{\left(\alpha ,\beta \right):\beta >0,\beta \le 2\alpha +1,\beta \le \alpha +1\right\}\\left\{\left(\alpha ,\beta \right):\alpha =0,\beta =1\right\}={E}_{7}\cup {E}_{8}\cup {E}_{9}.\end{array}$
1. (1)

We divide the proof of Theorem 1.1(1) into five cases.

Case 1.1 (α, β) E1 E2. From (1.1), (1.2), and applying

${\psi }^{m}\left(x\right)=\left(-{1\right)}^{m+1}{\int }_{0}^{\infty }\frac{{t}^{m}}{1-{e}^{-t}}{e}^{-xt}dt\phantom{\rule{0.5em}{0ex}}\left(x>0,\phantom{\rule{0.25em}{0ex}}m=1,2,...\right),$

we obtain for n ≥ 1,

$\begin{array}{l}\phantom{\rule{0.25em}{0ex}}{\left(-1\right)}^{n}{\left[\mathrm{log}{F}_{\alpha ,\beta }\left(x\right)\right]}^{\left(n\right)}\\ =\left(-{1\right)}^{n}\left[{\left(-1\right)}^{n-1}\frac{\alpha \left(n-1\right)!}{{x}^{n}}+\beta {\psi }^{\left(n-1\right)}\left(x\right)-{\beta }^{n}{\psi }^{\left(n-1\right)}\left(\beta x\right)\right]\\ =-\alpha {\int }_{0}^{\infty }{s}^{n-1}{e}^{-xs}ds+\beta {\int }_{0}^{\infty }\frac{{s}^{n-1}}{1-{e}^{-s}}{e}^{-xs}ds-{\beta }^{n}{\int }_{0}^{\infty }\frac{{t}^{n-1}}{1-{e}^{-t}}{e}^{-\beta xt}dt\\ =-\alpha {\beta }^{n}{\int }_{0}^{\infty }{t}^{n-1}{e}^{-\beta xt}dt+{\beta }^{n+1}{\int }_{0}^{\infty }\frac{{t}^{n-1}}{1-{e}^{-\beta t}}{e}^{-\beta xt}dt-{\beta }^{n}{\int }_{0}^{\infty }\frac{{t}^{n-1}}{1-{e}^{-t}}{e}^{-\beta xt}dt\\ ={\beta }^{n}{\int }_{0}^{\infty }\frac{{t}^{n-1}{e}^{-\beta xt}}{\left(1-{e}^{-t}\right)\left(1-{e}^{-\beta t}\right)}h\left(t\right)dt,\end{array}$
(3.1)

where

$h\left(t\right)=-\alpha {e}^{-\left(\beta +1\right)t}+\left(\alpha +1\right){e}^{-\beta t}+\left(\alpha -\beta \right){e}^{-t}-\alpha +\beta -1.$
(3.2)

Therefore, F α,β (x) is strictly logarithmically completely monotonic on (0, ∞), which follows from (3.1) and (3.2) together with Lemma 2.1(3).

Case 1.2 (α, β) E3. Then we clearly see that

${\left(-1\right)}^{n}{\left[log{F}_{\alpha ,\beta }\left(x\right)\right]}^{\left(n\right)}=\left(-{1\right)}^{n}\frac{\alpha \left(n-1\right)!\left(-{1\right)}^{n-1}}{{x}^{n}}=-\frac{\alpha \left(n-1\right)!}{{x}^{n}}>0$
(3.3)

for all x > 0.

Therefore, F α,β (x) is strictly logarithmically completely monotonic on (0, ∞), which follows from (3.3).

Case 1.3 (α,β) E4. Then F α,β (x) = 1 and

${\left(-1\right)}^{n}{\left[log{F}_{\alpha ,\beta }\left(x\right)\right]}^{\left(n\right)}=0.$
(3.4)

Therefore, F α,β (x) is not strictly logarithmically completely monotonic on (0, ∞), which follows from (3.4).

Case 1.4 (α, β) E5E6E7E8. Then F α,β (x) is not strictly logarithmically completely monotonic on (0, ∞), which follows from Lemmas 2.1(2), 2.1(4), 2.1(1), and equations (3.1) and (3.2).

Case 1.5 (α, β) E9. Then

${\left(-1\right)}^{n}{\left[log{F}_{\alpha ,\beta }\left(x\right)\right]}^{\left(n\right)}=-\frac{\alpha \left(n-1\right)!}{{x}^{n}}<0$
(3.5)

for all x > 0.

Therefore, F α,β (x) is not strictly logarithmically completely monotonic on (0, ∞), which follows from (3.5).

1. (2)

We divide the proof of Theorem 1.1(2) into five cases.

Case 2.1 (α, β) E7 E8. Then from (3.1) we get

$\begin{array}{c}\phantom{\rule{1em}{0ex}}{\left(-1\right)}^{n}{\left\{log{\left[{F}_{\alpha ,\beta }\left(x\right)\right]}^{-1}\right\}}^{\left(n\right)}\\ =-{\beta }^{n}{\int }_{0}^{\infty }\frac{{t}^{n-1}{e}^{-\beta xt}}{\left(1-{e}^{-t}\right)\left(1-{e}^{-\beta t}\right)}h\left(t\right)dt,\end{array}$
(3.6)

where

$h\left(t\right)=-\alpha {e}^{-\left(\beta +1\right)t}+\left(\alpha +1\right){e}^{-\beta t}+\left(\alpha -\beta \right){e}^{-t}-\alpha +\beta -1.$
(3.7)

Therefore, [F α,β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞), which follows from (3.6) and (3.7) together with Lemma 2.1(1).

Case 2.2 (α, β) E9. Then

${\left(-1\right)}^{n}{\left\{log{\left[{F}_{\alpha ,\beta }\left(x\right)\right]}^{-1}\right\}}^{\left(n\right)}=\frac{\alpha \left(n-1\right)!}{{x}^{n}}>0$
(3.8)

for all x > 0.

Therefore, [F α,β (x)]-1 is strictly logarithmically completely monotonic on (0, ∞), which follows from (3.8).

Case 2.3 (α, β) E4. Then [F α,β (x)]-1 = 1 and

${\left(-1\right)}^{n}{\left\{log{\left[{F}_{\alpha ,\beta }\left(x\right)\right]}^{-1}\right\}}^{\left(n\right)}=0.$
(3.9)

Therefore, [F α,β (x)]-1 is not strictly logarithmically completely monotonic on (0, ∞), which follows from (3.9).

Case 2.4 (α, β) E1 E2 E5 E6. Then [F α,β (x)]-1 is not strictly logarithmically completely monotonic on (0, ∞), which follows from equations (3.6) and (3.7) and Lemmas 2.1(3), 2.1(2) and 2.1(4).

Case 2.5 (α, β) E3. Then

${\left(-1\right)}^{n}{\left\{log{\left[{F}_{\alpha ,\beta }\left(x\right)\right]}^{-1}\right\}}^{\left(n\right)}=\frac{\alpha \left(n-1\right)!}{{x}^{n}}<0$
(3.10)

for all x > 0.

Therefore, [F α,β (x)]-1 is not strictly logarithmically completely monotonic on (0, ∞), which follows from (3.10). □

## References

1. Whittaker ET, Watson GN: A Course of Modern Analysis. Cambridge University Press, New York; 1962.

2. Chu Y-M, Zhang X-M, Zhang Zh-H: The geometric convexity of a function involving gamma function with applications. Commun Korean Math Soc 2010,25(3):373–383. 10.4134/CKMS.2010.25.3.373

3. Zhang X-M, Chu Y-M: A double inequality for gamma function. J Inequal Appl 2009, 2009: 1–7. Article ID 503782

4. Zhao T-H, Chu Y-M, Jiang Y-P: Monotonic and logarithmically convex properties of a function involving gamma functions. J Inequal Appl 2009, 2009: 1–13. Article ID 728612

5. Zhang X-M, Chu Y-M: An inequality involving the gamma function and the psi function. Int J Mod Math 2008,3(1):67–73.

6. Chu Y-M, Zhang X-M, Tang X-M: An elementary inequality for psi function. Bull Inst Math Acad Sin 2008,3(3):373–380.

7. Song Y-Q, Chu Y-M, Wu L-L: An elementary double inequality for gamma function. Int J Pure Math 2007,38(4):549–554.

8. Guo B-N, Qi F: Two new proofs of the complete monotonicity of a function involving the PSI function. Bull Korean Math Soc 2010,47(1):103–111. 10.4134/BKMS.2010.47.1.103

9. Chen Ch-P, Qi F, Srivastava HM: Some properties of functions related to the gamma and psi functions. Int Trans Spec Funct 2010,21(1–2):153–164.

10. Qi F: A completely monotonic function involving the divided difference of the psi function and an equivalent inequality involving sums. ANZIAM J 2007,48(4):523–532. 10.1017/S1446181100003199

11. Qi F, Guo B-N: Monotonicity and convexity of ratio between gamma functions to different powers. J Indones Math Soc 2005,11(1):39–49.

12. Chen Ch-P, Qi F: Inequalities relating to the gamma function. Aust J Math Anal Appl 2004,1(1):1–7. Article 3

13. Guo B-N, Qi F: Inequalities and monotonicity for the ratio of gamma functions. Taiwan J Math 2003,7(2):239–247.

14. Qi F: Monotonicity results and inequalities for the gamma and incomplete gamma functions. Math Inequal Appl 2002,5(1):61–67.

15. Qi F, Mei J-Q: Some inequalities of the incomplete gamma and related functions. Z Anal Anwendungen 1999,18(3):793–799.

16. Qi F, Guo S-L: Inequalities for the incomplete gamma and related functions. Math Inequal Appl 1999,2(1):47–53.

17. Alzer H: Some gamma function inequalities. Math Comp 1993,60(201):337–346. 10.1090/S0025-5718-1993-1149288-7

18. Alzer H: On some inequalities for the gamma and psi functions. Math Comp 1997,66(217):373–389. 10.1090/S0025-5718-97-00807-7

19. Anderson GD, Qiu S-L: A monotonicity property of the gamma function. Proc Am Math Soc 1997,125(11):3355–3362. 10.1090/S0002-9939-97-04152-X

20. Kershaw D: Some extensions of W. Gautschi's inequalities for the gamma function. Math Comp 1983,41(164):607–611.

21. Merkle M: Logarithmic convexity and inequalities for the gamma function. J Math Anal Appl 1996,203(2):369–380. 10.1006/jmaa.1996.0385

22. Palumbo B: A generalization of some inequalities for the gamma function. J Comput Appl Math 1998,88(2):255–268. 10.1016/S0377-0427(97)00187-8

23. Alzer H, Berg Ch: Some classes of completely monotonic functions II. Ramanujan J 2006,11(2):225–248. 10.1007/s11139-006-6510-5

24. Alzer H: Sharp inequalities for the digamma and polygamma functions. Forum Math 2004,16(2):181–221. 10.1515/form.2004.009

25. Alzer H, Batir N: Monotonicity properties of the gamma function. Appl Math Lett 2007,20(7):778–781. 10.1016/j.aml.2006.08.026

26. Clark WE, Ismail MEH: Inequalities involving gamma and psi functions. Anal Appl 2003,1(1):129–140. 10.1142/S0219530503000041

27. Elbert Á, Laforgia A: On some properties of the gamma function. Proc Am Math Soc 2000,128(9):2267–2673.

28. Bustoz J, Ismail MEH: On gamma function inequalities. Math Comp 1986,47(176):659–667. 10.1090/S0025-5718-1986-0856710-6

29. Ismail MEH, Lorch L, Muldoon ME: Completely monotonic functions associated with the gamma function and its q-analogues. J Math Anal Appl 1986,116(1):1–9. 10.1016/0022-247X(86)90042-9

30. Babenko VF, Skorokhodov DS: On Kolmogorov-type inequalities for functions defined on a semiaxis. Ukr Math J 2007,59(10):1299–1312.

31. Muldoon ME: Some monotonicity properties and characterizations of the gamma function. Aequationes Math 1978,18(1–2):54–63. 10.1007/BF01844067

32. Qi F, Yang Q, Li W: Two logarithmically completely monotonic functions connected with gamma function. Int Trans Spec Funct 2006,17(7):539–542. 10.1080/10652460500422379

33. Qi F, Niu D-W, Cao J: Logarithmically completely monotonic functions involving gamma and polygamma functions. J Math Anal Approx Theory 2006,1(1):66–74.

34. Qi F, Chen Sh-X, Cheung W-S: Logarithmically completely monotonic functions concerning gamma and digamma functions. Int Trans Spec Funct 2007,18(6):435–443. 10.1080/10652460701318418

35. Qi F: A class of logarithmically completely monotonic functions and the best bounds in the first Kershaw's double inequality. J Comput Appl Math 2007,206(2):1007–1014. 10.1016/j.cam.2006.09.005

36. Chen Ch-P, Qi F: Logarithmically complete monotonicity properties for the gamma function. Aust J Math Anal Appl 2005,2(2):1–9. Article 8

37. Chen Ch-P, Qi F: Logarithmically completely monotonic functions relating to the gamma function. J Math Anal Appl 2006,321(1):405–411. 10.1016/j.jmaa.2005.08.056

38. Merkle M: On log-convexity of a ratio of gamma functions. Univ Beograd Publ Elektrotehn Fak Ser Mat 1997, 8: 114–119.

39. Chen Ch-P: Complete monotonicity properties for a ratio of gamma functions. Univ Beograd Publ Elektrotehn Fak Ser Mat 2005, 16: 26–28.

40. Li A-J, Chen Ch-P: Some completely monotonic functions involving the gamma and polygamma functions. J Korean Math Soc 2008,45(1):273–287. 10.4134/JKMS.2008.45.1.273

## Acknowledgements

This study is partly supported by the Natural Science Foundation of China (Grant no. 11071069), the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant no. T200924), and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant no. Y200805602).

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Correspondence to Yu-Ming Chu.

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

Y-PL carried out the proof of Theorem 1.1 in this paper. T-CS carried out the proof of Lemma 2.1 in this paper. Y-MC provided the main idea of this paper. All authors read and approved the final manuscript.

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Lv, YP., Sun, TC. & Chu, YM. Necessary and sufficient conditions for a class of functions and their reciprocals to be logarithmically completely monotonic. J Inequal Appl 2011, 36 (2011). https://doi.org/10.1186/1029-242X-2011-36 