access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present some monotonic functions and some generalized inequalities involving the ratios of analogues of the Gamma function. Mathematics Subject Classification: 33B15, 26A4

Copyright C K. Nantomah

E. Prempeh

K. Nantomah

M. M. Iddrisu

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Int. Journal of Math. Analysis, Vol. 8, 2014, no. 18, 895 - 900
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ijma.2014.4390
Generalization of Some Inequalities
for the Ratio of Gamma Functions
K. Nantomah and M. M. Iddrisu
Department of Mathematics
University for Development Studies
Navrongo Campus, P. O. Box 24
Navrongo, UE/R, Ghana
E. Prempeh
Department of Mathematics
Kwame Nkrumah University of Science and Technology
Kumasi, Ghana
Copyright c   2014 K. Nantomah, M. M. Iddrisu and E. Prempeh. This is an open
access article distributed under the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited.
Abstract
We present some monotonic functions and some generalized inequalities involving
the ratios of analogues of the Gamma function.
Mathematics Subject Classiﬁcation: 33B15, 26A48
Keywords: Gamma Function, p-analogue, q-analogue, k-analogue, In-
equality
1 Introduction
The classical Euler’s Gamma function Γ(t) is commonly deﬁned as
Γ(t)=
 ∞
0
e
−xx
t−1 dx, t > 0. (1)896 K. Nantomah, M. M. Iddrisu and E. Prempeh
The p-digamma function ψp(t), q-digamma function ψq(t) and k-digamma
function ψk(t) are respectively deﬁned as follows.
ψp(t)=
d
dt
ln(Γp(t)) =
Γ 
p(t)
Γp(t)
,t > 0. (2)
where Γp(t) is the p-analogue of the Gamma function deﬁned by (see [2], [3])
Γp(t)=
p!pt
t(t +1 )...(t + p)
=
pt
t(1 + t
1)...(1 + t
p)
,p ∈ N, t > 0, (3)
ψq(t)=
d
dt
ln(Γq(t)) =
Γ 
q(t)
Γq(t)
,t > 0 (4)
where Γq(t) is the q-analogue of the Gamma function deﬁned by (see [4])
Γq(t)=( 1− q)
1−t
∞ 
n=1
1 − qn
1 − qt+n,q ∈ (0,1),t > 0, (5)
and
ψk(t)=
d
dt
ln(Γk(t)) =
Γ 
k(t)
Γk(t)
,t > 0 (6)
where Γk(t) is the k-analogue of the Gamma function deﬁned by (see [1], [5])
Γk(t)=
 ∞
0
e
− xk
k x
t−1 dx, k > 0,t > 0. (7)
In a recent paper [6], Nantomah and Iddrisu proved that the following double
inequalities hold:
0 <
Γk(α + t)
Γp(α + t)
<
e(t−1)(
ln k−γ
k )Γk(α +1 )
tpt−1Γp(α +1 )
,k > 0,p∈ N (8)
and
0 <
Γk(α + t)
Γq(α + t)
<
e(−t)(
ln k−γ
k )Γk(α +1 )
t(1 − q)1−tΓq(α +1 )
,k > 0,q∈ (0,1) (9)
for t ∈ (0,1) and for a positive real number α.
Our objective in this paper is to establish some generalizations of the inequal-
ities (8) and (9).Generalization of some inequalities 897
2 Preliminary Results
The following auxiliary results are crucial to the main results of the paper.
Lemma 2.1. The functions ψp(t), ψq(t) and ψk(t) as deﬁned above have the
following series representations.
ψp(t)=l np −
p 
n=0
1
n + t
,p ∈ N, t > 0 (10)
ψq(t)=−ln(1 − q)+l nq
∞ 
n=0
qt+n
1 − qt+n,q ∈ (0,1),t > 0 (11)
ψk(t)=
lnk − γ
k
−
1
t
+
∞ 
n=1
t
nk(nk + t)
,k > 0,t > 0. (12)
where γ is the Euler-Mascheroni’s constant.
Proof. See [3], [5] and [6] and the references therein.
Lemma 2.2. Let a>0, b>0 and t>0. Then,
−a(
lnk − γ
k
)+blnp +
a
t
+ aψk(t) − bψp(t) > 0.
Proof. Using the series representations in equations (10) and (12) we have,
− a(
lnk − γ
k
)+blnp +
a
t
+ aψk(t) − bψp(t)
= a
∞ 
n=1
t
nk(nk + t)
+ b
p 
n=0
1
(n + t)
> 0.
Lemma 2.3. Let a>0, b>0 and α + βt > 0. Then
−a(
lnk − γ
k
)+blnp +
a
t
+ aψk(α + βt) − bψp(α + βt) > 0
Proof. This follows directly from Lemma 2.2.
Lemma 2.4. Let a>0, b>0 and t>0. Then,
−a(
lnk − γ
k
)+bln(1 − q)+
a
t
+ aψk(t) − bψq(t) > 0898 K. Nantomah, M. M. Iddrisu and E. Prempeh
Proof. Using the series representations in equations (11) and (12) we have,
− a(
lnk − γ
k
)+bln(1 − q)+
a
t
+ aψk(t) − bψq(t)
= a
∞ 
n=1
t
nk(nk + t)
− blnq
∞ 
n=0
qt+n
1 − qt+n > 0
Lemma 2.5. Let a>0, b>0 and α + βt > 0. Then,
−a(
lnk − γ
k
)+bln(1 − q)+
a
t
+ aψk(α + βt) − bψp(α + βt) > 0
Proof. This follows directly from Lemma 2.4.
3 Main Results
We now state and prove the results of this paper.
Theorem 3.1. Deﬁne a function Λ by
Λ(t)=
taβe−aβt(
ln k−γ
k )Γk(α + βt)a
p−bβtΓp(α + βt)b ,t ∈ (0,∞),k>0,p∈ N. (13)
where a, b, α, β are positive real numbers. Then Λ is increasing on t ∈ (0,∞)
and the inequality
0 <
Γk(α + βt)a
Γp(α + βt)b <
eaβ(t−1)(
ln k−γ
k )Γk(α + β)a
taβpbβ(t−1)Γp(α + β)b (14)
holds for every t ∈ (0,1).
Proof. Let g(t) = lnΛ(t) for every t ∈ (0,∞). Then,
g(t)=l n
taβe−aβt(
ln k−γ
k )Γk(α + βt)a
p−bβtΓp(α + βt)b
= −aβt(
lnk − γ
k
)+bβtlnp + aβ lnt + alnΓk(α + βt) − blnΓp(α + βt)
Then,
g
 (t)=−aβ(
lnk − γ
k
)+bβ lnp +
aβ
t
+ aβψk(α + βt) − bβψp(α + βt)
= β

−a(
lnk − γ
k
)+blnp +
a
t
+ aψk(α + βt) − bψp(α + βt)

> 0Generalization of some inequalities 899
as a result of Lemma 2.3. This proves that g is increasing on t ∈ (0,∞). Hence
Λ is increasing on t ∈ (0,∞). Thus, for every t ∈ (0,1) we have
Λ(0) < Λ(t) < Λ(1),
yielding
0 <
Γk(α + βt)a
Γp(α + βt)b <
eaβ(t−1)(
ln k−γ
k )Γk(α + β)a
taβpbβ(t−1)Γp(α + β)b .
Corollary 3.2. If t ∈ [1,∞), then the following inequality holds.
eaβ(t−1)(
ln k−γ
k )Γk(α + β)a
taβpbβ(t−1)Γp(α + β)b ≤
Γk(α + βt)a
Γp(α + βt)b
Proof. If t ∈ [1,∞), then we have Λ(1) ≤ Λ(t) yielding the result.
Theorem 3.3. Deﬁne a function Υ by
Υ(t)=
taβe−aβt(
ln k−γ
k )Γk(α + βt)a
(1 − q)bβtΓq(α + βt)b ,t ∈ (0,∞),k>0,q∈ (0,1). (15)
where a, b, α, β are positive real numbers. Then Υ is increasing on t ∈ (0,∞)
and the inequality
0 <
Γk(α + βt)a
Γq(α + βt)b <
eaβ(t−1)(
ln k−γ
k )Γk(α + β)a
taβ(1 − q)bβ(1−t)Γq(α + β)b (16)
holds for every t ∈ (0,1).
Proof. Let h(t) = lnΥ(t) for every t ∈ (0,∞). Then,
h(t)=l n
taβe−aβt(
ln k−γ
k )Γk(α + βt)a
(1 − q)bβtΓq(α + βt)b
= −aβt(
lnk − γ
k
) − bβtln(1 − q)+aβ lnt + alnΓk(α + βt) − blnΓq(α + βt)
Then,
h
 (t)=−aβ(
lnk − γ
k
) − bβ ln(1 − q)+
aβ
t
+ aβψk(α + βt) − bβψq(α + βt)
= β

−a(
lnk − γ
k
) − bln(1 − q)+
a
t
+ aψk(α + βt) − bψp(α + βt)

> 0900 K. Nantomah, M. M. Iddrisu and E. Prempeh
as a result of Lemma 2.5. This proves that h is increasing on t ∈ (0,∞). Hence
Υ is increasing on t ∈ (0,∞) and for every t ∈ (0,1) we have
Υ(0) < Υ(t) < Υ(1)
yielding
0 <
Γk(α + βt)a
Γq(α + βt)b <
eaβ(t−1)(
ln k−γ
k )Γk(α + β)a
taβ(1 − q)bβ(1−t)Γq(α + β)b.
Corollary 3.4. If t ∈ [1,∞), then the following inequality holds.
eaβ(t−1)(
ln k−γ
k )Γk(α + β)a
taβ(1 − q)bβ(1−t)Γq(α + β)b ≤
Γk(α + βt)a
Γq(α + βt)b
Proof. If t ∈ [1,∞), then we have Υ(1) ≤ Υ(t) yielding the result.
4 Concluding Remarks
Remark 4.1. By putting a = b = β = 1 into inequalities (14) and (16), we
thus obtain respectively, inequalities (8) and (9) as in [6].
References
[1] R. D´ iaz and E. Pariguan, On hypergeometric functions and Pachhammer
k-symbol, Divulgaciones Matemt´ icas 15(2)(2007), 179-192.
[2] V. Krasniqi, T. Mansour and A. Sh. Shabani, Some Monotonicity Prop-
erties and Inequalities for Γ and ζ Functions, Mathematical Communica-
tions 15(2)(2010), 365-376.
[3] V. Krasniqi and F. Merovci, Logarithmically completely monotonic
functions involving the generalized gamma function, Le Matematiche
LXV(2010), 15-23.
[4] T. Mansour, Some inequalities for the q-Gamma Function, J. Ineq. Pure
Appl. Math. 9(1)(2008), Art. 18.
[5] F. Merovci, Power Product Inequalities for the Γk Function, Int. Journal
of Math. Analysis 4(21)(2010), 1007-1012.
[6] K. Nantomah and M. M. Iddrisu, Some Inequalities Involving the Ratio of
Gamma Functions, Int. Journal of Math. Analysis 8(12)(2014), 555-560.
Received: March 19, 2014

Generalization of Some Inequalities for the Ratio of Gamma Functions

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Generalization of Some Inequalities for the Ratio of Gamma Functions

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