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Some Analytical Properties of the Remainder of Binet-Like Expression for the Barnes

G-Function

Emmanuel Ansong Adjei, Kwara Nantomah∗, Morgan Yindobil Zubil
Department of Mathematics, School of Mathematical Sciences, C. K. Tedam University of Technology and

Applied Sciences, P. O. Box 24, Navrongo, Upper-East Region, Ghana
adjeid2@gmail.com, knantomah@cktutas.edu.gh, morganzubil@gmail.com

∗Correspondence: knantomah@cktutas.edu.gh

Abstract. In this paper, we prove some properties such as monotonicity, complete monotonicity, log-arithmic convexity, inequalities, subadditivity and starshapedness, involving the Binet-like remainderof the Barnes G-function. The methods of proofs are analytical in nature.

1. Introduction
Special functions are usually encountered in almost every scientific discipline. Particularly, theyplay important roles in areas such as mathematics, physics and engineering. The gamma function,which is an extension of the factorial function, is arguably the most important special function. Thisis largely due to its vast areas of applications as well as its connection with other special functions.It is usually defined as

Γ(z) =

∫ ∞
0

tz−1e−tdt (1)
for z > 0. The Binet’s formula for logarithm of the gamma function is given as

ln Γ(z) =

(
z −

1

2

)
ln z − z + ln

√
2π + θ(z) (2)

for z > 0, where
θ(z) =

∫ ∞
0

(
1

et − 1
−

1

t
+

1

2

)
e−zt

t
dt (3)

is known as the remainder of Binet’s formula. Due to the important properties exhibited by thefunction θ(z), it has been investigated in multiple ways. In [5], the authors proved among otherthings that, for p ∈ (0, 1], the function
fp(z) = θ(pz)− pθ(z) (4)

Received: 20 Dec 2024.2010 Mathematics Subject Classification. 33B15, 26A48, 26A51.
Key words and phrases. Multiple gamma function; Barnes G-function; Binet Remainder; Binet-like remainder; log-convex function; completely monotonic function; subadditive function; starshaped function.1

https://adac.ee
https://doi.org/10.28924/ada/ma.5.13


Eur. J. Math. Anal. 10.28924/ada/ma.5.13 2is completely monotonic on (0,∞). In [7], the authors investigated the complete monotonicity ofthe function
fp,q,r (z) = r [θ(pz)− qθ(z)] (5)

where p > 0, q is a real number and r 6= 0. They further established that θ(z) is subadditiveon (0,∞) and −θ(z) is starshaped on (0,∞). In [11], the authors considered a generalization ofthe function θ(z) which is denoted by θα(z). They investigated the complete monotonicity of thefunction
fp,q,α(z) = θα(pz)− qθα(z) (6)

where p > 0, q is a real number and α > 0. Among other things, they further established that thefunction θα(z) is subadditive. In [3], the authors considered an inequality for the r-th derivative of
θ(z). Subsequently, they obtain the Turan-type inequality(

θ(r+1)(z)
)2
≤ θ(r)(z)θ(r+2)(z) (7)

where r ∈ N. A generalization of (7) can be found in [2].The multiple gamma function, which is a generalization of the ordinary gamma function, wasdefined by Barnes as [1]
Γr+1(z + 1) =

Γr+1(z + 1)

Γr (z)
, z ∈ C, r ∈ N,

Γ1(z) = Γ(z),

Γr (1) = 1.

The particular case G(z) = 1
Γ2(z) is referred to as the Barnes G-function or the double gammafunction. It satisfies the following basic properties [1].

G(z + 1) = G(z)Γ(z), z ∈ C,

G(1) = 1,

(lnG(z))′′′ ≥ 0, z > 0.

For further properties of the function G(z), one may refer to the papers [10] and [12] and thereferences in there.A Binet-like expression for logarithm of the double gamma function is given by Choi [6] as
ln Γ2(z) = lnA−

z2

4
+

(
z2

2
−
z

2
+

1

12

)
ln z + (1− z) ln Γ(z) + Θ(z) (8)

where A = 1.282427... is the Glaisher-Kinkelin constant and
Θ(z) =

∫ ∞
0

(
1

t
−

1

et − 1
−

1

2
+

t

12

)
e−zt

t2
dt. (9)

is what is referred to as the Binet-like remainder.

https://doi.org/10.28924/ada/ma.5.13


Eur. J. Math. Anal. 10.28924/ada/ma.5.13 3It is interesting to observe that the Binet-like remainder, Θ(z) has some resemblance withthe Binet remainder, θ(z). The natural question that arise is: does the function Θ(z) satisfythe properties satisfied by the function θ(z) ? Motivated by the papers [3, 5, 7, 11], the objectiveof this paper is to answer this question. We establish some properties of the function Θ(z)such as monotonicity, complete monotonicity, logarithmic convexity, inequalities, subadditivity andstarshapedness, among others. We present our findings in the next section. Before that, we providethe following definitions which shall pave the way for us to prove our results. Throughout thispaper, N = {1, 2, 3, . . . } and N0 = {0, 1, 2, 3, . . . }.
Definition 1.1. A real-valued function K defined on an interval I ⊆ R is said to be convex on I iff

K
(x
u

+
y

v

)
≤
K(x)

u
+
K(y)

v
(10)

holds for all x, y ∈ I and u > 1, v > 1 such that 1
u + 1

v = 1. Equivalently, K is said to be convexon I iff
K′′(z) ≥ 0 (11)

for all z ∈ I . If the inequalities (10) and (11) are reversed, then K is said to be concave on I .
Definition 1.2. A positive real-valued function K defined on an interval I ⊆ R is said to belogarithmically convex on I iff

K
(x
u

+
y

v

)
≤ [K(x)]

1
u [K(y)]

1
v (12)

holds for all x, y ∈ I and u > 1, v > 1 such that 1
u + 1

v = 1. Equivalently, K is said to belogarithmically convex on I iff
[lnK(z)]′′ ≥ 0 (13)

for all z ∈ I . If the inequalities (12) and (13) are reversed, then K is said to be logarithmicallyconcave on I .
Definition 1.3 ( [13]). A real-valued function K defined on an interval I ⊆ R is said to be completelymonotonic on I iff

(−1)nK(n)(z) ≥ 0

holds for all z ∈ I and n ∈ N0.
Definition 1.4 ( [4]). A real-valued function K defined on an interval I ⊆ R is said to be subadditiveon I iff

K(x + y) ≤ K(x) +K(y)

holds for all x, y ∈ I . If the inequality is reversed, then K is said to be superadditive on I .

https://doi.org/10.28924/ada/ma.5.13


Eur. J. Math. Anal. 10.28924/ada/ma.5.13 4

Definition 1.5 ( [4]). A real-valued function K defined on an interval I ⊆ R is said to be starshapedon I iff
K(αz) ≤ αK(z),for all z ∈ I and α ∈ [0, 1].

2. Results and Discussion
Beginning with the following lemmas, we now present our findings in this section.

Lemma 2.1. For t > 0, the inequality
1

t2
−

1

12
<

e−t

(1− e−t)2
<

1

t2
(14)

holds.

Proof. See Theorem 2 of [8]. �

Lemma 2.2. For t > 0, the function

P(t) =
1

t2
−

e−t

(1− e−t)2
(15)

is strictly decreasing.

Proof. See Theorem 1 of [8] or Theorem 1.1 of [9]. �

Theorem 2.3. For t > 0, let A(t) be defined as

A(t) =
1

t
−

1

et − 1
−

1

2
+

t

12
. (16)

=
t

12
+

1

t
−

1

2
coth(

t

2
). (17)

Then:

(a) A(t) is strictly increasing.
(b) A(t) is positive.
(c) A(t) is strictly convex.

Proof. By L’hopital’s rule and simple computation, we have
lim
t→0

A(t) = 0 and lim
t→∞

A(t) =∞.

Then by making use of the left-hand side of (14), we obtain
A′(t) =

et

(et − 1)2
−

1

t2
+

1

12

=
e−t

(e−t − 1)2
−

1

t2
+

1

12

> 0.

https://doi.org/10.28924/ada/ma.5.13


Eur. J. Math. Anal. 10.28924/ada/ma.5.13 5Hence A(t) is strictly increasing and that completes the proof of (a). Next, the increasing propertyof A(t) implies that for t > 0,
A(t) > lim

t→0
A(t) = 0which completes the proof of (b). Next, as a result of Lemma 2.2, we have

A′′(t) = −
(

1

t2
−

e−t

(1− e−t)2

)′
= −P ′(t) > 0

which completes the proof of (c). �

Remark 2.4. Theorem 2.3 (c) implies that Θ(z) is positive.
Corollary 2.5. For t > 0, the inequality

e−t

(1− e−t)2
<

2

t3
(18)

holds.

Proof. The convexity of A(t) implies that
A′′(t) =

2

t3
+

et

(et − 1)2
−

2e2t

(et − 1)3
> 0.

This simplifies to
2

t3
(et − 1)3 − et(et − 1) > 0which further simplifies to

2

t3
>

et

(et − 1)2
=

e−t

(1− e−t)2
.

This completes the proof. �

Remark 2.6. We note that 1
t2 − 2

t3 < 0 if 0 < t < 2 and 1
t2 − 2

t3 > 0 if t > 2. Thus, the upperbound in (14) is better that upper bound in (18) if 0 < t < 2 and the upper bound in (18) is betterthan the upper bound in (14) if t > 2.
Theorem 2.7. The function Θ(z) is completely monotonic on (0,∞).

Proof. Differentiating r number of times of (9) gives
Θ(r)(z) = (−1)r

∫ ∞
0

A(t)tr−2e−ztdt (19)
where r ∈ N0 and Θ(0)(z) = Θ(z). This implies that

(−1)rΘ(r)(z) = (−1)2r

∫ ∞
0

A(t)tr−2e−ztdt

=

∫ ∞
0

A(t)tr−2e−ztdt > 0.

This completes the proof. �

https://doi.org/10.28924/ada/ma.5.13


Eur. J. Math. Anal. 10.28924/ada/ma.5.13 6

Remark 2.8. The representation (19) implies that the function Θ(z) is decreasing and convex on
(0,∞).
Theorem 2.9. Let r ∈ N0 be even. Then the function Θ(r)(z) is logarithmically convex on (0,∞).
That is, the inequality

Θ(r)
(x
u

+
y

v

)
≤
[

Θ(r)(x)
] 1
u
[

Θ(r)(y)
] 1
v (20)

holds for x > 0, y > 0, u > 1, v > 1 and 1
u + 1

v = 1.

Proof. Let r ∈ N0 be an even number. Then, by using (19) and Holder’s inequality for integrals,we obtain
Θ(r)

(x
u

+
y

v

)
=

∫ ∞
0

A(t)tr−2e−( x
u

+ y
v

)tdt

=

∫ ∞
0

(A(t)tr−2)
1
u

+ 1
v e−( x

u
+ y
v

)tdt

=

∫ ∞
0

A(t)
1
u t

r−2
u e

−xt
u A(t)

1
v t

r−2
v e

−yt
v dt

≤
(∫ ∞

0

A(t)tr−2e−xtdt

) 1
u
(∫ ∞

0

A(t)tr−2e−ytdt

) 1
v

=
[

Θ(r)(x)
] 1
u
[

Θ(r)(y)
] 1
v

which completes the proof. �

Remark 2.10. The particular case where r = 0 in Theorem 2.9 proves that Θ(z) is logarithmicallyconvex on (0,∞).
Remark 2.11. Theorem 2.9 shows that, for even r ∈ N0, the function

T (z) =
Θ(r+1)(z)

Θ(r)(z)
(21)

is increasing on (0,∞).
Corollary 2.12. Let r ∈ N0 be even and p ∈ (0, 1]. Then the function

B(z) =
Θ(r)(pz)

[Θ(r)(z)]p

decreasing on (0,∞). Consequently, for 0 < x ≤ y , the inequality(
Θ(r)(y)

Θ(r)(x)

)p
≥

Θ(r)(py)

Θ(r)(px)
(22)

holds.

https://doi.org/10.28924/ada/ma.5.13


Eur. J. Math. Anal. 10.28924/ada/ma.5.13 7

Proof. Logarithmic differentiation of B(z) and applying the increasing property of T (z) gives
B′(z)

B(z)
= p

Θ(r+1)(pz)

Θ(r)(pz)
− p

Θ(r+1)(z)

Θ(r)(z)

= p

[
Θ(r+1)(pz)

Θ(r)(pz)
−

Θ(r+1)(z)

Θ(r)(z)

]
≤ 0.

Hence B(z) is decreasing. Consequently, for 0 < x ≤ y , we have B(x) ≥ B(y) which whenrearranged gives (22). �

Remark 2.13. If p ≥ 1 in Corollary 2.12, then the reverse cases of the conclusions are obtained.
Theorem 2.14. Let r ∈ N0 and s ∈ N0. Then the inequality∣∣∣Θ( r

u
+ s
v

)
(x
u

+
y

v

)∣∣∣ ≤ ∣∣∣Θ(r)(x)
∣∣∣ 1
u
∣∣∣Θ(s)(y)

∣∣∣ 1
v (23)

holds for x > 0, y > 0, u > 1, v > 1 and 1
u + 1

v = 1.

Proof. By using (19) and Holder’s inequality for integrals, we obtain∣∣∣Θ( r
u

+ s
v

)
(x
u

+
y

v

)∣∣∣ =

∫ ∞
0

A(t)t( r
u

+ s
v

)−2e−( x
u

+ y
v

)tdt

=

∫ ∞
0

A(t)( 1
u

+ 1
v

)t( r
u

+ s
v

)−2( 1
u

+ 1
v

)e−( x
u

+ y
v

)tdt

=

∫ ∞
0

A(t)
1
u t

r−2
u e

−xt
u A(t)

1
v t

s−2
v e

−yt
v dt

≤
(∫ ∞

0

A(t)tr−2e−xtdt

) 1
u
(∫ ∞

0

A(t)ts−2e−ytdt

) 1
v

=
[

Θ(r)(x)
] 1
u
[

Θ(s)(y)
] 1
v

which completes the proof. �

Remark 2.15. If r = s in Theorem 2.14, then we obtain∣∣∣Θ(r)
(x
u

+
y

v

)∣∣∣ ≤ ∣∣∣Θ(r)(x)
∣∣∣ 1
u
∣∣∣Θ(r)(y)

∣∣∣ 1
v (24)

which shows that, for r ∈ N0, the function ∣∣Θ(r)(z)
∣∣ is logarithmically convex on (0,∞).

Remark 2.16. If r = k − 1, s = k + 1, u = v = 2 and x = y = z in Theorem 2.14, then we obtainthe Turan-type inequality ∣∣∣Θ(k) (z)
∣∣∣2 ≤ ∣∣∣Θ(k−1)(z)

∣∣∣ ∣∣∣Θ(k+1)(z)
∣∣∣ . (25)

where k ∈ N0. This is equivalent to∣∣∣Θ(k+1) (z)
∣∣∣2 ≤ ∣∣∣Θ(k)(z)

∣∣∣ ∣∣∣Θ(k+2)(z)
∣∣∣ (26)

https://doi.org/10.28924/ada/ma.5.13


Eur. J. Math. Anal. 10.28924/ada/ma.5.13 8where k ∈ N0.
Lemma 2.17. The function Θ′(z) is increasing on (0,∞).

Proof. This follows directly from (19) since (Θ′(z))′ = Θ′′(z) > 0. �

Theorem 2.18. The function −Θ(z) is starshaped on (0,∞). That is, the inequality

−Θ(αz) ≤ −αΘ(z) (27)
holds for α ∈ [0, 1] and z ∈ (0,∞).

Proof. Let K(z) = Θ(αz)− αΘ(z) for α ∈ [0, 1] and z ∈ (0,∞). Then
K′(z) = αΘ′(αz)− αΘ′(z)

= α
[
Θ′(αz)−Θ′(z)

]
< 0

since Θ′(z) is increasing. Hence K(z) is decreasing. Then for z ∈ (0,∞), we have
K(z) ≥ lim

z→∞
K(z) = 0

which implies that
Θ(αz) ≥ αΘ(z).This then gives rise to the inequality (27) and that completes the proof. �

Theorem 2.19. Let r ∈ N0. Then Θ(r)(z) is strictly subadditive if r is even and Θ(r)(z) is strictly
superadditive if r is odd. That is, for x, y ∈ (0,∞), it holds that

Θ(r)(x + y) < Θ(r)(x) + Θ(r)(y) (28)
if r is even, and

Θ(r)(x + y) > Θ(r)(x) + Θ(r)(y) (29)
if r is odd.

Proof. Let U(x, y) = Θ(r)(x + y) −Θ(r)(x) −Θ(r)(y). With no loss of generality, let y be fixed.Then by differentiating with respect to x , and using (19), we have
U ′(x, y) = Θ(r+1)(x + y)−Θ(r+1)(x)

(−1)r+1

∫ ∞
0

A(t)tr−1e−(x+y)tdt − (−1)r+1

∫ ∞
0

A(t)tr−1e−xtdt

= (−1)r+1

∫ ∞
0

A(t)tr−1
[
e−(x+y)t − e−xt

]
dt

:= V(x, y)

https://doi.org/10.28924/ada/ma.5.13


Eur. J. Math. Anal. 10.28924/ada/ma.5.13 9Suppose that r is even. Then V(x, y) > 0. This implies that U(x, y) is increasing in terms of x .Hence, for x ∈ (0,∞), we have
U(x, y) < lim

x→∞
U(x, y) = −Θ(r)(y) < 0

which gives rise to the inequality (28). Likewise, suppose that r is odd. Then V(x, y) < 0. Thisimplies that U(x, y) is decreasing in terms of x . Hence, for x ∈ (0,∞), we have
U(x, y) > lim

x→∞
U(x, y) = −Θ(r)(y) > 0

which gives rise to the inequality (29). This completes the proof. �

Remark 2.20. The particular case where r = 0 in Theorem 2.19, shows that the function Θ(z) isstrictly subadditive on (0,∞).
References

[1] V. S. Adamchik, Contributions to the Theory of the Barnes Function, Int. J. Math. Comput. Sci. 9 (2014), 11-30.
https://future-in-tech.net/9.1/R-AdamchikContributions.pdf.[2] T. Batbold, Some Remarks on Results of Mortici, Kragujevac J. Math. 36 (2012), 73-76. https://imi.pmf.kg.ac.
rs/kjm/pub/13476258401367_kjom3601-08.pdf.[3] P. K. Bhandari and S. K. Bissu, On some inequalities involving Turan-type inequalities, Cogent Math. 3 (2016),1130678. https://doi.org/10.1080/23311835.2015.1130678.[4] A. M. Bruckner and E. Ostrow, Some function classes related to the class of convex functions, Pac. J. Math. 12(1962), 1203-1215. https://doi.org/10.2140/PJM.1962.12.1203.[5] C-P. Chen and F. Qi, Completely Monotonic Functions Related to the Gamma Functions, RGMIA Res. Rep. Coll. 8(2005), 3. https://rgmia.org/papers/v8n2/Comp-monoto-property.pdf.[6] J. Choi, Some mathematical constants, Appl. Math. Comput. 187 (2007), 122-140. https://doi.org/10.1016/j.
amc.2006.08.091.[7] S. Guo and F. Qi, A Class of Completely Monotonic Functions Related to the Remainder of Binet’s Formula withApplications, Tamsui Oxf. J. Math. Sci. 25 (2009), 9-14. https://ibi.au.edu.tw/var/file/18/1018/img/1838/
25(1)-2-009.pdf.[8] A-Q. Liu, G-F. Li, B-N. Guo and F. Qi, Monotonicity and logarithmic concavity of two functions involving exponentialfunction, Int. J. Math. Ed. Sci. Tech. 39 (2008), 686-691. https://doi.org/10.1080/00207390801986841.[9] K. Nantomah, Alternative proof of a monotonicity property of certain function, Math. Anal. Contemp. Appl. 5 (2023),65-68. https://doi.org/10.30495/maca.2023.1987128.1067.[10] Y. A. Neretin, The double gamma function and Vladimir Alekseevsky, arXiv:2402.07740. https://arxiv.org/pdf/
2402.07740.[11] F. Qi and B-N. Guo, Some Properties of Extended Remainder of Binet’s First Formula for Logarithm of GammaFunction, Math. Slovaca, 60 (2010), 461-470. https://doi.org/10.2478/s12175-010-0025-7.[12] S. Sandow, K. Nantomah and B. Seidu, On Some Analytical Properties of the Barnes G-Function, Asia Pac. J.Math. 8 (2021), 7. https://apjm.apacific.org/PDFs/8-7.pdf.[13] D. V. Widder, The Laplace Transform, Princeton University Press, London, 1941. https://archive.org/details/
dli.ernet.206074.

https://doi.org/10.28924/ada/ma.5.13
https://future-in-tech.net/9.1/R-AdamchikContributions.pdf
https://imi.pmf.kg.ac.rs/kjm/pub/13476258401367_kjom3601-08.pdf
https://imi.pmf.kg.ac.rs/kjm/pub/13476258401367_kjom3601-08.pdf
https://doi.org/10.1080/23311835.2015.1130678
https://doi.org/10.2140/PJM.1962.12.1203
https://rgmia.org/papers/v8n2/Comp-monoto-property.pdf
https://doi.org/10.1016/j.amc.2006.08.091
https://doi.org/10.1016/j.amc.2006.08.091
https://ibi.au.edu.tw/var/file/18/1018/img/1838/25(1)-2-009.pdf
https://ibi.au.edu.tw/var/file/18/1018/img/1838/25(1)-2-009.pdf
https://doi.org/10.1080/00207390801986841
https://doi.org/10.30495/maca.2023.1987128.1067
https://arxiv.org/pdf/2402.07740
https://arxiv.org/pdf/2402.07740
https://doi.org/10.2478/s12175-010-0025-7
https://apjm.apacific.org/PDFs/8-7.pdf
https://archive.org/details/dli.ernet.206074
https://archive.org/details/dli.ernet.206074

	1. Introduction
	2. Results and Discussion
	References

