The function 1/x ln Γ(x+1)−ln x+1 is strictly completely monotonic

on (0,∞)

Chen, Chao-Ping

Qi, Feng

English

AbstractA logarithmically completely monotonic function is completely monotonic. The function 1−lnx+1xlnΓ(x+1) is strictly completely monotonic on (0,∞). The function Γ(x+1)x/x is strictly logarithmically completely monotonic on (0,∞)

Elsevier - Publisher Connector 

J. Math. Anal. Appl. 296 (2004) 603–607
www.elsevier.com/locate/jmaa
A complete monotonicity property
of the gamma function ✩
Feng Qi, Chao-Ping Chen ∗
Department of Applied Mathematics and Informatics, Jiaozuo Institute of Technology, Jiaozuo City,
Henan 454000, China
Received 20 July 2003
Available online 2 July 2004
Submitted by H.M. Srivastava
Abstract
A logarithmically completely monotonic function is completely monotonic. The function 1 −
lnx + 1x lnΓ (x + 1) is strictly completely monotonic on (0,∞). The function x
√
Γ (x + 1)/x is
strictly logarithmically completely monotonic on (0,∞).
 2004 Elsevier Inc. All rights reserved.
Keywords: Gamma function; Psi function; Logarithmically completely monotonic function
1. Introduction
The classical gamma function is usually defined for Re z > 0 by
Γ (z) =
∞∫
0
tz−1e−t dt. (1)
✩ The authors were supported in part by NSF (no. 10001016) of China, SF for the Prominent Youth of Henan
Province (no. 0112000200), SF of Henan Innovation Talents at Universities, Doctor Fund of Jiaozuo Institute of
Technology, China.
* Corresponding author.
E-mail addresses: qifeng@jzit.edu.cn, fengqi618@member.ams.org (F. Qi), chenchaoping@jzit.edu.cn
(C.-P. Chen).
URL: http://rgmia.vu.edu.au/qi.html, http://dami.jzit.edu.cn/staff/qifeng.html (F. Qi).0022-247X/$ – see front matter  2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jmaa.2004.04.026
604 F. Qi, C.-P. Chen / J. Math. Anal. Appl. 296 (2004) 603–607The psi or digamma function, the logarithmic derivative of the gamma function, and the
polygamma functions can be expressed [6, p. 16] as
ψ(x) = Γ
′(x)
Γ (x)
= −γ +
∞∫
0
e−t − e−xt
1 − e−t dt, (2)
ψ(m)(x) = (−1)m+1
∞∫
0
tm
1 − e−t e
−xt dt (3)
for x > 0 and m ∈N, where γ = 0.57721566490153286 . . . is the Euler–Mascheroni con-
stant.
In 1985, D. Kershaw and A. Laforgia [5] showed that the function x[Γ (1 + 1/x)]x is
strictly increasing on (0,∞), which is equivalent to the function [Γ (x + 1)]1/x/x being
strictly decreasing on (0,∞). In addition, it was proved that the function x1−γ [Γ (1 +
1/x)x] decreases for 0 < x < 1, which is equivalent to [Γ (1 + x)]1/x/x1−γ being increas-
ing on (1,∞).
In [2,10], it is proved that the function f (x) = [Γ (x + 1)]1/x/(x + 1) is strictly
decreasing and strictly logarithmically convex in (0,∞) and the function g(x) =
[Γ (x + 1)]1/x/√x + 1 is strictly increasing and strictly logarithmically concave in (0,∞).
Some new proofs for the monotonicity of the function xr [Γ (x +1)]1/x on (0,∞) are given
for r /∈ (0,1). In addition, if s is a positive real number, then for all real numbers x > 0,
e−γ
[Γ (s + 1)]1/s <
[Γ (x + 1)]1/x
[Γ (x + s + 1)]1/(x+s) < 1, (4)
limx→0 f (x) = e−γ and limx→∞ f (x) = e−1.
Using monotonicity and inequalities of the generalized weighted mean values [1,
7,8,12], the first author proved [9] that the functions [Γ (s)/Γ (r)]1/(s−r), [Γ (s, x)/
Γ (r, x)]1/(s−r), and [γ (s, x)/γ (r, x)]1/(s−r) are increasing in r > 0, s > 0, and x > 0.
For any given x > 0, the function (sγ (s, x))/xs is decreasing in s > 0.
In [3], N. Elezovic´, C. Giordana, and J. Pecˇaric´, among others, verified the convexity
with respect to variable x of the function [Γ (x + t)/Γ (x + s)]1/(t−s) for |t − s| < 1.
Recall that a function f is said to be completely monotonic on an interval I if f has
derivatives of all orders on I which alternate successively in sign, that is
(−1)nf (n)(x) 0 (5)
for x ∈ I and n 0. If inequality (5) is strict for all x ∈ I and for all n 0, then f is said
to be strictly completely monotonic.
Similarly, we give the following definition.
Definition 1. A function f is said to be logarithmically completely monotonic on an inter-
val I if its logarithm lnf satisfies
(−1)k[lnf (x)](k)  0 (6)
for k ∈N on I . If inequality (6) is strict for all x ∈ I and for all k  1, then f is said to be
strictly logarithmically completely monotonic.
F. Qi, C.-P. Chen / J. Math. Anal. Appl. 296 (2004) 603–607 605In [13] it was established that the function 1 + 1
x
lnΓ (x + 1) − ln(x + 1) is strictly
completely monotone in (−1,∞) and tends to 1 as x → −1 and to 0 as x → ∞. This
property is derived from the following integral representation:
lnΓ (x + 1) = x ln(x + 1) − x +
∞∫
0
(
1
t
− 1
et − 1
)
e−t 1 − e
−xt
t
dt. (7)
In this short note, we are about to prove a complete monotonicity result of a function
involving the gamma function as follows.
Theorem 1. A (strictly) logarithmically completely monotonic function is also (strictly)
completely monotonic.
Theorem 2. The function 1 − lnx + 1
x
lnΓ (x + 1) is strictly completely monotonic on
(0,∞) and tends to ∞ as x → 0 and to 0 as x → ∞. Moreover, the function x√Γ (x + 1)/x
is strictly logarithmically completely monotonic on (0,∞).
2. Proofs of theorems
Proof of Theorem 1. It is clear that expφ(x)  0. Further, it is easy to see that
[expφ(x)]′ = φ′(x) expφ(x) 0 and [expφ(x)]′′ = {φ′′(x) + [φ′(x)]2} expφ(x) 0.
Suppose (−1)k[expφ(x)](k)  0 for all nonnegative integers k  n, where n ∈ N is a
given positive integer. By Leibnitz’s formula, we have
(−1)n+1[expφ(x)](n+1) = (−1)n+1{[expφ(x)]′}(n) = (−1)n+1[φ′(x) expφ(x)](n)
= (−1)n+1
n∑
i=0
(
n
i
)
φ(i+1)(x)
[
expφ(x)
](n−i)
=
n∑
i=0
(
n
i
)[
(−1)i+1φ(i+1)(x)]{(−1)n−i[expφ(x)](n−i)}
 0. (8)
By induction, it is proved that the function expφ(x) is completely monotonic. 
Proof of Theorem 2. It has been shown in [5] that the function [Γ (x + 1)]1/x/x is strictly
decreasing on (0,∞), then f (x) = 1− lnx+ 1
x
lnΓ (x+1) is strictly decreasing on (0,∞).
From the asymptotic expansion in [4]:
lnΓ (x) =
(
x − 1
2
)
lnx − x + ln√2π + 1
12x
+ O(x−3) as x → ∞, (9)
we conclude that limx→∞ f (x) = 0 and limx→0 f (x) = ∞. This implies f (x) > 0 for
x > 0.
606 F. Qi, C.-P. Chen / J. Math. Anal. Appl. 296 (2004) 603–607Using Leibnitz’ rule
[
u(x)v(x)
](n) =
n∑
k=0
(
n
k
)
u(k)(x)v(n−k)(x), (10)
we obtain
f (n)(x) =
n∑
k=0
(
n
k
)(
1
x
)(n−k)[
lnΓ (x + 1)](k) − (−1)n−1(n − 1)!
xn
=
(
1
x
)(n)
lnΓ (x + 1)+
n∑
k=1
(
n
k
)(
1
x
)(n−k)
ψ(k−1)(x + 1)+ (−1)
nn!
nxn
= (−1)
nn!
xn+1
lnΓ (x + 1) +
n∑
k=1
n!
k!
(−1)n−k
xn−k+1
ψ(k−1)(x + 1) + (−1)
nn!
nxn
= (−1)n n!
xn+1
g(x), (11)
and
g′(x) = (−1)
n
n! x
nψ(n)(x + 1) + 1
n
. (12)
Using (3) and (n − 1)!/xn = ∫∞0 tn−1e−xt dt for x > 0 and n ∈N, we conclude
1
xn
g′(x) = 1
n!
∞∫
0
(
1 − t
et − 1
)
tn−1e−xt dt > 0, (13)
since 0 < t/(et − 1) < 1 for t > 0. Thus, the function g is strictly increasing and g(x) >
g(0) = 0 on (0,∞), which implies (−1)nf (n)(x) > 0 for x > 0 and n = 0,1,2, . . . .
The rest follows from Theorem 1. The proof is complete. 
Remark. It is worthwhile to point out that the integral form and completely monotonic
property of the function (bt − at)/t for t ∈ (−∞,∞) have been researched in [11]. The
function (bt − at)/t is important to the extended mean values [7,8].
Acknowledgment
The authors thank the anonymous referee for his/her many valuable suggestions to improve this manuscript.
References
[1] B.-N. Guo, F. Qi, Inequalities for generalized weighted mean values of convex function, Math. Inequal.
Appl. 4 (2) (2001) 195–202, RGMIA Res. Rep. Coll. 2 (7) (1999) 1059–1065, Art. 11. Available online at
http://rgmia.vu.edu.au/v2n7.html.
F. Qi, C.-P. Chen / J. Math. Anal. Appl. 296 (2004) 603–607 607[2] Ch.-P. Chen, F. Qi, Monotonicity results for the gamma function, J. Inequal. Pure Appl. Math. 3 (2003), Art.
44. Available online at http://jipam.vu.edu.au/article.php?sid=282. RGMIA Res. Rep. Coll. 5 (2002), Art.
16. Available online at http://rgmia.vu.edu.au/v5(E).html.
[3] N. Elezovic´, C. Giordano, J. Pecˇaric´, The best bounds in Gautschi’s inequality, Math. Inequal. Appl. 3 (2000)
239–252.
[4] C.L. Frenzen, Error bounds for asymptotic expansions of the ratio of two gamma functions, SIAM J. Math.
Anal. 18 (1987) 890–896.
[5] D. Kershaw, A. Laforgia, Monotonicity results for the gamma function, Atti Accad. Sci. Torino Cl. Sci. Fis.
Mat. Natur. 119 (1985) 127–133.
[6] W. Magnus, F. Oberhettinger, R.P. Soni, Formulas and Theorems for the Special Functions of Mathematical
Physics, Springer-Verlag, Berlin, 1966.
[7] F. Qi, Generalized abstracted mean values, J. Inequal. Pure Appl. Math. 1 (2000), Art. 4. Available online at
http://jipam.vu.edu.au/article.php?sid=97. RGMIA Res. Rep. Coll. 2 (5) (1999) 633–642, Art. 4. Available
online at http://rgmia.vu.edu.au/v2n5.html.
[8] F. Qi, Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng.
Sci. 454 (1978) (1998) 2723–2732.
[9] F. Qi, Monotonicity results and inequalities for the gamma and incomplete gamma functions, Math. Inequal.
Appl. 5 (1) (2002) 61–67. RGMIA Res. Rep. Coll. 2 (7) (1999) 1027–1034, Art. 7. Available online at
http://rgmia.vu.edu.au/v2n7.html.
[10] F. Qi, Ch.-P. Chen, Monotonicity and convexity results for functions involving the gamma function, RGMIA
Res. Rep. Coll. 6 (4) (2003), Art. 10. Available online at http://rgmia.vu.edu.au/v6n4.html.
[11] F. Qi, S.-L. Xu, The function (bx − ax)/x: Inequalities and properties, Proc. Amer. Math. Soc. 126 (11)
(1998) 3355–3359.
[12] F. Qi, Sh.-Q. Zhang, Note on monotonicity of generalized weighted mean values, R. Soc. Lond. Proc. Ser. A
Math. Phys. Eng. Sci. 455 (1989) (1999) 3259–3260.
[13] H. Vogt, J. Voigt, A monotonicity property of the Γ -function, J. Inequal. Pure Appl. Math. 3 (5) (2002), Art.
73. Available online at http://jipam.vu.edu.au/article.php?sid=255.


A complete monotonicity property of the gamma function 

Victoria University Eprints Repository

A COMPLETE MONOTONICITY OF THE GAMMA FUNCTIONFENG QI AND CHAO-PING CHENAbstract. The function 1xln Γ(x+1)−lnx+1 is strictly completely monotonicon (0,∞).The classical gamma function is usually defined for Re z > 0 byΓ(z) =∫ ∞0tz−1e−t dt. (1)The psi or digamma function, the logarithmic derivative of the gamma function,and the polygamma functions can be expressed [6, p. 16] asψ(x) =Γ′(x)Γ(x)= −γ +∫ ∞0e−t − e−xt1− e−t dt, (2)ψ(m)(x) = (−1)m+1∫ ∞0tm1− e−t e−xt dt (3)for x > 0 and m ∈ N, where γ = 0.57721566490153286 · · · is the Euler-Mascheroniconstant.In 1985, D. Kershaw and A. Laforgia [5] showed that the function x[Γ(1 + 1x )]xis strictly increasing on (0,∞), which is equivalent to the function [Γ(x+1)]1/xx beingstrictly decreasing on (0,∞). In addition, it was proved that the function x1−γ [Γ(1+1x )x] decreases for 0 < x < 1, which is equivalent to [Γ(1+x)]1xx1−γ being increasing on(1,∞).In [2, 10], it is proved that the function f(x) = [Γ(x+1)]1/xx+1 is strictly decreasingand strictly logarithmically convex in (0,∞) and the function g(x) = [Γ(x+1)]1/x√x+1isstrictly increasing and strictly logarithmically concave in (0,∞). Some new proofsfor the monotonicity of the funtion xr[Γ(x+1)]1/x on (0,∞) are given for r 6∈ (0, 1).In addition, if s is a positive real number, then for all real numbers x > 0,e−γ[Γ(s+ 1)]1/s<[Γ(x+ 1)]1/x[Γ(x+ s+ 1)]1/(x+s)< 1, (4)limx→0 f(x) = e−γ and limx→∞ f(x) = e−1.Using monotonicity and inequalities of the generalized weighted mean values [1,7, 8, 12], the first author proved [9] that the functions[Γ(s)Γ(r)]1/(s−r),[Γ(s,x)Γ(r,x)]1/(s−r)2000 Mathematics Subject Classification. 33B15.Key words and phrases. Gamma function, psi function, completely monotonic function.The authors were supported in part by NNSF (#10001016) of CHINA, SF for the ProminentYouth of Henan Province (#0112000200), SF of Henan Innovation Talents at Universities, DoctorFund of Jiaozuo Institute of Technology, CHINA.12 F. QI AND CH.-P. CHENand[γ(s,x)γ(r,x)]1/(s−r)are increasing in r > 0, s > 0 and x > 0. For any given x > 0,the function sγ(s,x)xs is decreasing in s > 0.In [3], N. Elezovic´, C. Giordana and J. Pecˇaric´, among others, verified the con-vexity with respect to variable x of the function[Γ(x+t)Γ(x+s)]1/(t−s)for |t− s| < 1.Recall that a function f is said to be completely monotonic on an interval I if fhas derivatives of all orders on I which alternate successively in sign, that is(−1)nfn(x) ≥ 0 (5)for x ∈ I and n ≥ 0. If inequality (5) is strict for all x ∈ I and for all n ≥ 0, thenf is said to be strictly completely monotonic. See [11] and references therein.In this short note, we are about to prove a complete monotonicity result of afunction involving the gamma function.Theorem 1. The function 1x ln Γ(x+1)− lnx+1 is strictly completely monotonicon (0,∞) and tends to ∞ as x→ 0 and to 0 as x→∞.Proof. It has been shown in [5] that the function [Γ(x+1)]1/xx is strictly decreasingon (0,∞), then f(x) = 1x ln Γ(x+1)− lnx+1 is strictly decreasing on (0,∞). Fromthe asymptotic expansion in [4]:ln Γ(x) =(x− 12)lnx− x+ ln√2pi +112x+O(x−3) as x→∞,we conclude that limx→∞ f(x) = 0 and limx→0 f(x) = ∞. This implies f(x) > 0for x > 0.Using Leibniz’ rule[u(x)v(x)](n) =n∑k=0(nk)u(k)(x)v(n−k)(x) (6)we obtainf (n)(x) =n∑k=0(nk)(1x)(n−k)[ln Γ(x+ 1)](k) − (−1)n−1(n− 1)!xn=(1x)(n)ln Γ(x+ 1) +n∑k=1(nk)(1x)(n−k)ψ(k−1)(x+ 1) +(−1)nn!nxn=(−1)nn!xn+1ln Γ(x+ 1) +n∑k=1n!k!(−1)n−kxn−k+1ψ(k−1)(x+ 1) +(−1)nn!nxn, (−1)n n!xn+1g(x),(7)andg′(x) =(−1)nn!xnψ(n)(x+ 1) +1n. (8)Using (3) and (n−1)!xn =∫∞0tn−1e−xt dt for x > 0 and n ∈ N, we conclude1xng′(x) =1n!∫ ∞0(1− tet − 1)tn−1e−xt dt > 0, (9)A COMPLETE MONOTONICITY OF THE GAMMA FUNCTION 3since 0 < tet−1 < 1 for x > 0. Thus, the function g is strictly increasing and g(x) >g(0) = 0 on (0,∞), which implies (−1)nf (n)(x) > 0 for x > 0 and n = 0, 1, 2, . . . .The proof is complete. Remark 1. After this paper was finalized, a similar result in [13] was found: Thefunction 1 + 1x ln Γ(x + 1) − ln(x + 1) is strictly completely monotone in (−1,∞)and tends to 1 as x → −1 and to 0 as x → ∞. This property is derived from thefollowing integral representation:ln Γ(x+ 1) = x ln(x+ 1)− x+∫ ∞0(1t− 1et − 1)e−t1− e−xttdt. (10)References[1] B.-N. Guo and F. Qi, Inequalities for generalized weighted mean values of convex function,Math. Inequal. Appl. 4 (2001), no. 2, 195–202. RGMIA Res. Rep. Coll. 2 (1999), no. 7,Art. 11, 1059–1065. Available online at http://rgmia.vu.edu.au/v2n7.html.[2] Ch.-P. Chen and F. Qi, Monotonicity results for the gamma function, J. Inequal. Pure Appl.Math. 3 (2003), no. 2, Art. 44. Available online at http://jipam.vu.edu.au/v4n2/065_02.html. RGMIA Res. Rep. Coll. 5 (2002), suppl., Art. 16. Available online at http://rgmia.vu.edu.au/v5(E).html.[3] N. Elezovic´, C. Giordano and J. Pecˇaric´, The best bounds in Gautschi’s inequality, Math.Inequal. Appl. 3 (2000), 239–252.[4] C. L. Frenzen, Error bounds for asymptonic expansions of the ratio of two gamma functions,SIAM J. Math. 18 (1987), 890–896.[5] D. Kershaw and A. Laforgia, Monotonicity results for the gamma function, Atti Accad. Sci.Torino Cl. Sci. Fis. Mat. Natur. 119 (1985), 127–133.[6] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Func-tions of Mathematical Physics, Springer, Berlin, 1966.[7] F. Qi, Generalized abstracted mean values, J. Inequal. Pure Appl. Math. 1 (2000), no. 1,Art. 4. Available online at http://jipam.vu.edu.au/v1n1/013_99.html. RGMIA Res. Rep.Coll. 2 (1999), no. 5, Art. 4, 633–642. Available online at http://rgmia.vu.edu.au/v2n5.html.[8] F. Qi, Generalized weighted mean values with two parameters, R. Soc. Lond. Proc. Ser. AMath. Phys. Eng. Sci. 454 (1998), no. 1978, 2723–2732.[9] F. Qi, Monotonicity results and inequalities for the gamma and incomplete gamma functions,Math. Inequal. Appl. 5 (2002), no. 1, 61–67. RGMIA Res. Rep. Coll. 2 (1999), no. 7, Art. 7,1027–1034. Available online at http://rgmia.vu.edu.au/v2n7.html.[10] F. Qi and Ch.-P. Chen,Monotonicity and convexity results for functions involving the gammafunction, RGMIA Res. Rep. Coll. 6 (2003), no. 4, Art. 10. Available online at http://rgmia.vu.edu.au/v6n4.html.[11] F. Qi and S.-L. Xu, The function (bx−ax)/x: Inequalities and properties, Proc. Amer. Math.Soc. 126 (1998), no. 11, 3355–3359.[12] F. Qi and Sh.-Q. Zhang, Note on monotonicity of generalized weighted mean values, R. Soc.Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999), no. 1989, 3259–3260.[13] H. Vogt and J. Voigt, A monotonicity property of the Γ-function, J. Inequal. Pure Appl. Math.3 (2002), no. 5, Art. 73. Available online at http://jipam.vu.edu.au/v3n5/007_01.html.(F. Qi) Department of Applied Mathematics and Informatics, Jiaozuo Institute ofTechnology, Jiaozuo City, Henan 454000, CHINAE-mail address: qifeng@jzit.edu.cn, fengqi618@member.ams.orgURL: http://rgmia.vu.edu.au/qi.html(Ch.-P. Chen) Department of Applied Mathematics and Informatics, Jiaozuo Instituteof Technology, Jiaozuo City, Henan 454000, CHINAE-mail address: chenchaoping@sohu.com

A monotonicity property of the Γ-function,

cari´ c, The best bounds in Gautschi’s inequality,

Error bounds for asymptonic expansions of the ratio of two gamma functions,

Formulas and Theorems for the Special Functions of

Generalized abstracted mean values,

Generalized weighted mean values with two parameters,

Inequalities for generalized weighted mean values of convex function,

Monotonicity and convexity results for functions involving the gamma function,

Monotonicity results and inequalities for the gamma and incomplete gamma functions,

Monotonicity results for the gamma function,

Note on monotonicity of generalized weighted mean values,

The function (bx−ax)/x: Inequalities and properties,

A Complete Monotonicity of the Gamma Function

The function 1/x ln Γ(x+1)−ln x+1 is strictly completely monotonic\ud
on (0,∞)

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A Complete Monotonicity of the Gamma Function

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