1. No category

Turkish Journal of Analysis and Number Theory, 2014, Vol. 2, No. 6, 226-229
Available online at http://pubs.sciepub.com/tjant/2/6/7
© Science and Education Publishing
DOI:10.12691/tjant-2-6-7
Generalized Inequalities Related to the Classical Euler’s
Gamma Function
Kwara Nantomah*
Department of Mathematics, University for Development Studies, Navrongo Campus, Navrongo UE/R, Ghana
*Corresponding author: [email protected], [email protected]
Received October 10, 2014; Revised November 26, 2014; Accepted December 14, 2014
Abstract This paper presents some inequalities concerning certain ratios of the classical Euler’s Gamma function.
The results generalized some recent results.
Keywords: Gamma function, q-Gamma function, k-Gamma function, (p,q)-Gamma function, (q,k)-Gamma
function, inequality
Cite This Article: Kwara Nantomah, “Generalized Inequalities Related to the Classical Euler’s Gamma
Function.” Turkish Journal of Analysis and Number Theory, vol. 2, no. 6 (2014): 226-229. doi: 10.12691/tjant-26-7.
is the k-generalized Pochhammer symbol.
Furthermore, Krasniqi and Merovci [4] defined the
(p,q)-Gamma function Γ( p,q ) (t ) for p ∈ N , q ∈ (0,1) and
1. Introduction
We begin by outlining the following basic definitions
well-known in literature.
The celebrated classical Euler’s Gamma function, Γ(t )
is defined for t > 0 as
∞
t > 0 as
[ p ]tq [ p ]q !
Γ( p,q ) (t ) =
,
[t ]q [t + 1]q [t + p ]q
where
− x t −1
Γ(t ) =
∫ e x dx.
[ p ]q =
0
The q-Gamma function, Γ q (t ) is defined for q ∈ (0,1)
and t > 0 as (see [2])
∞
Γ q (t ) =
(1 − q )1−t ∏
1 − qn
n =1 1 − q
t +n
.
Also, the k-Gamma function, Γ k (t ) was defined by
Diaz and Pariguan [1] for k > 0 and t > 0 as
∞
k
− x t −1
k x dx.
Γ k (t ) =
∫e
1− q p
.
1− q
The psi function, ψ (t ) otherwise known as the
digamma function is defined as the logarithmic derivative
of the Gamma function. That is,
d
Γ′(t )
InΓ(t )=
.
Γ(t )
dt
ψ (t )=
The q-digamma function, k-digamma function, (p,q)diagamma function and (q,k)-digamma function are
similarly defined as follows:
0
Diaz and Teruel [5] further defined the (q,k)-Gamma
function Γ( q,k ) (t ) for q ∈ (0,1) , k > 0 and t > 0 as
t −1
(1 − q k )qk,k
,
Γ( q,k ) (t ) =
ψ q (t ) =
Γ′q (t )
d
InΓ q (t ) =
,
dt
Γ q (t )
ψ k (t ) =
Γ′ (t )
d
InΓ k (t ) = k ,
dt
Γ k (t )
d
dt
Γ′( p,q ) (t )
ψ ( p,q ) (t ) = InΓ( p,q ) (t ) =
t −1
(1 − q ) k
Γ( p,q ) (t )
and
where
(t )n,k = t (t + k )(t + 2k ) (t + (n − 1)k )
=
n −1
∏ (t + jk )
j =0
d
dt
Γ′( q,k ) (t )
ψ ( q,k ) (t ) = InΓ( q,k ) (t ) =
Γ( q,k ) (t )
.
It is common knowledge that these functions exhibit the
following series charaterizations (see also [7-12]):
Turkish Journal of Analysis and Number Theory
∞
ψ q (t ) =
−In(1 − q ) + (Inq ) ∑
q nt
n =1 1 − q
n
∞
t
Ink - γ 1
− +∑
k
t n =1 nk (nk + t )
ψ k=
(t )
p
ψ ( p=
, q ) (t ) In[ p ]q + (Inq ) ∑
q nt
n =1 1 − q
n
∞
q nkt
-In(1 − q )
+ (Inq ) ∑
ψ=
( q , k ) (t )
nk
k
n =1 1 − q
(1)
(2)
where γ = 0.577215664901532... represents the EulerMascheroni’s constant.
Of late, the following double inequalities were
presented in [7] by the use of some monotonicity
properties of some functions related with the Gamma
function.
(1 − q )−t Γ q (α )
[ p ]tq Γ( p,q ) (α )
≥
≥
Γ( p,q ) (α + t )
(1 − q )1−t Γ q (α + 1)
[ p ]tq−1 Γ( p,q ) (α + 1)
≥
Γ( q,k ) (α )
1 (1−t )
(1 − q ) k
t −γ t
kΓ
Γ( q,k ) (α + t )
(6)
<
p
 ∞ q nt
q nt 
= (Inq ) u ∑
− w∑
 ≤ 0.
n
− qn
 n 1 1=
n 1 1 − q 
=
We conclude the proof by substituting t by α + β t .
Lemma 2.2. Suppose that α > 0 , β > 0
k (α
k (α )
<
Γ( q,k ) (α )
t −γ t
αk k e k Γ
(7)
+ 1)
(α + t )(1 − q )
Γ k (α + t )
Γ( q,k ) (α + t )
In(1- q )
k
+ uψ q (α + β t ) − wψ ( q,k ) (α + β t ) ≤ 0.
uIn(1 − q ) − w
In(1- q )
+ uψ q (t ) − wψ ( q,k ) (t )
k
∞ 
q nt
q nkt 
= (Inq ) ∑ u
−w
 ≤ 0.
n
1 − q nk 
n =1  1 − q
Ink uγ
u
+
+
k
k α + βt
+ uψ k (α + β t ) − wψ ( p,q ) (α + β t ) > 0.
1−t
k Γ
Ink uγ u
+
+ + uψ q (t ) − wψ ( p,q ) (t )
k
k t
∞
p
t
q nt
− w∑
> 0.
n
nk (nk + t )
n 11 − q
1=
= u∑
n
=
We conclude the proof by substituting t by α + β t .
Lemma 2.4. Suppose that α > 0 , β > 0 , u > 0 w > 0 ,
t > 0 , q ∈ (0,1) and k > 0 . Then,
+ 1)
( q , k ) (α
Proof. From the characterization in equations (2) and (3)
we obtain,
wIn[ p ]q − u
(8)
k (α
,
wIn[ p ]q − u
(α + t )[ p ]tq−1 Γ( p,q ) (α + 1)
t −γ t
αk k e k Γ
<
uIn(1 − q ) + wIn[ p]q + uψ q (t ) − wψ ( p,q ) (t )
t > 0 , k > 0 , p ∈ N and q ∈ (0,1) . Then,
for t ∈ (0,1) , α > 0 , p ∈ N , q ∈ (0,1) and k > 0 .
(α
Proof. From the characterization in equations (1) and (3)
we obtain,
k (α )
t −1 γ (1−t )
+ 1)k k e k Γ
−t
+ t )(1 − q ) k
+ uψ q (α + β t ) − wψ ( p,q ) (α + β t ) ≤ 0.
We conclude the proof by substituting t by α + β t .
Lemma 2.3. Suppose that α > 0 , β > 0 , u > 0 w > 0 ,
Γ( q,k ) (α + 1)
Γ k (α + t )
<
t
(α + t )[ p ]q Γ( p,q ) (α ) Γ( p,q ) (α + t )
(α
uIn(1 − q ) + wIn[ p]q
uIn(1 − q ) − w
for t ∈ (0,1) , α > 0 , q ∈ (0,1) and k ≥ 1 .
αk k e
β >0 ,
Proof. From the characterization in equations (1) and (4)
we obtain,
Γ q (α + t )
(1 − q )1−t Γ q (α + 1)
≥
,
u ≥ w > 0 , t > 0 , p ∈ N and q ∈ (0,1) . Then,
(5)
for t ∈ (0,1) , α > 0 , p ∈ N and q ∈ (0,1) .
−t
(1 − q ) k
We begin with the following Lemmas.
Lemma 2.1. Suppose that α > 0
u ≥ w > 0 , t > 0 , q ∈ (0,1) and k ≥ 1 . Then,
Γ q (α + t )
(1 − q )−t Γ q (α )
2. Supporting Results
(3)
(4)
227
+ 1)
for t ∈ (0,1) , α > 0 , q ∈ (0,1) and k > 0 .
Results of this form can also be found in [8,9,10,11,12].
By utilizing similar techniques as in the previous results,
this paper seeks to provide some generalizations of the
above inequalities. We present our results in the following
sections.
In(k u (1 − q ) w ) uγ
u
+
+
k
k α + βt
+ uψ k (α + β t ) − wψ ( q,k ) (α + β t ) > 0.
−
Proof. From the characterization in equations (2) and (4)
we obtain,
228
−
Turkish Journal of Analysis and Number Theory
(1 − q )−u β t Γ q (α )u
In(k u (1 − q ) w ) uγ
u
+
+
+ uψ q (t ) − wψ ( q,k ) (t )
k
k α + βt
∞
∞
t
q
− w(Inq ) ∑
> 0.
nk
nk (nk + t )
1=
n 11 − q
=
u∑
n
(1 − q )
nkt
−
wβ t
k Γ
≥
We conclude the proof by substituting t by α + β t .
η (t ) = In
q ∈ (0,1) by
[ p ]−q wβ t Γ( p,q ) (α + β t ) w
(9)
, t ∈ (0, ∞)
u ≥ w . Then, E is non-increasing on t ∈ (0, ∞) and the
inequalities:
−u β t
Γ q (α )
wβ t
[ p ]q Γ( p,q ) (α ) w
Γ q (α + β t )
u
≥
u
Γ( p,q ) (α + β t ) w
(1 − q )u β (1−t ) Γ q (α + β )u
(10)
[ p ]qwβ (t −1) Γ( p,q ) (α + β ) w
are valid for each t ∈ (0,1) .
Proof. Let λ (t ) = InE (t ) for every t ∈ (0, ∞) . Then
(1 − q )u β t Γ q (α + β t )u
wβ
(1−t )
k
Γ
( q , k ) (α
+ β )w
wβ
In(1 − q )
k
+ u βψ q (α + β t ) − wβψ ( q,k ) (α + β t )
η ′=
(t ) u β In(1 − q ) −
In(1 − q )

= β uIn(1 − q ) − w
k

+ uψ q (α + β t ) − wψ ( q,k ) (α + β t )  ≤ 0
as a result of Lemma 2.2. That implies η is nonincreasing on t ∈ (0, ∞) . Consequently, F is nonincreasing on t ∈ (0, ∞) and for each t ∈ (0,1) we have,
G (t ) =
+ u βψ q (α + β t ) − wβψ ( p,q ) (α + β t )
= β uIn(1 − q ) + wIn[ p ]q
as a result of Lemma 2.1. That implies λ is nonincreasing on t ∈ (0, ∞) . Consequently, E is nonincreasing on t ∈ (0, ∞) and for each t ∈ (0,1) we have,
α
u
−
u β t u βγ t
k e k Γ
k (α
+ β t )u
(13)
[ p ]−q wβ t Γ( p,q ) (α + β t ) w
u β t u βγ t
−
k k e k Γ
yielding equation (10).
Theorem 3.2. Define a function F for q ∈ (0,1) and
k ≥ 1 by
(11)
<
(α + β )
u
u
<
Γ k (α + β t )u
Γ( p,q ) (α + β t ) w
u β (t −1) u βγ (1−t )
k k e k Γ
(α + β t )
u
k (α + β )
wβ (t −1)
[ p ]q
Γ( p,q ) (α + β ) w
(14)
u
are valid for each t ∈ (0,1) .
Proof. Let µ (t ) = InG (t ) for every t ∈ (0, ∞) . Then
w
where u , w , α , β are positive real numbers such that
u ≥ w . Then, F is non-increasing on t ∈ (0, ∞) and the
inequalities:
k (α )
(α + β t )u [ p ]qwβ t Γ( p,q ) (α ) w
E (0) ≥ E (t ) ≥ E (1)
+ β t)
(α + β t )u k
where u , w , α , β are positive real numbers. Then, G
is increasing on t ∈ (0, ∞) and the inequalities:
+ uψ q (α + β t ) − wψ ( p,q ) (α + β t )  ≤ 0
( q ,k ) (α
+ β t )w
Then,
(t ) u β In(1 − q ) + wβ In[ p]q
λ ′=
, t ∈ (0, ∞)
( q ,k ) (α
yielding equation (12).
Theorem 3.3. Define a function G for t ∈ (0, ∞) ,
p ∈ N , q ∈ (0,1) and k > 0 by
Then,
(1 − q )u β t Γ q (α + β t )u
uβ t
k Γ
F (0) ≥ F (t ) ≥ F (1)
+ uInΓ q (α + β t ) − wInΓ( p,q ) (α + β t )
uβ t
(1 − q ) k Γ
(1 − q )u β t Γ q (α + β t )u
wβ t
In(1 − q )
k
+ uInΓ q (α + β t ) − wInΓ( q,k ) (α + β t )
[ p ]q− wβ t Γ( p,q ) (α + β t ) w
= u β tIn(1 − q ) + wβ tIn[ p ]q
=
F (t )
(12)
= u β tIn(1 − q ) −
where u , w , α , β are positive real numbers such that
λ (t ) = In
Γ( q,k ) (α + β t ) w
(1 − q )u β (1−t ) Γ q (α + β )u
(1 − q )
(1 − q )u β t Γ q (α + β t )u
≥
w
Γ q (α + β t )u
are valid for each t ∈ (0,1) .
Proof. Let η (t ) = InF (t ) for every t ∈ (0, ∞) . Then
We now present our results in the following Theorems.
Theorem 3.1. Define a function E for p ∈ N and
(1 − q )
( q , k ) (α )
(1 − q )
3. Main Results
E (t )
=
≥
µ (t ) = In
(α + β t )u k
−
u β t u βγ t
k e k Γ
k (α
+ β t )u
[ p ]−q wβ t Γ( p,q ) (α + β t ) w
Turkish Journal of Analysis and Number Theory
uβ t
u βγ t
Ink +
+ wβ tIn[ p ]q
k
k
+ uInΓ k (α + β t ) − wInΓ( p,q ) (α + β t ).
229
In(k u (1 − q ) w ) u βγ
uβ
+
+
α + βt
k
k
+ u βψ k (α + β t ) − wβψ ( q,k ) (α + β t )
= uIn(α + β t ) −
−β
δ ′(t ) =
 In(k u (1 − q ) w ) uγ
u
= β −
+
+
α
βt
+
k
k

Then,
Ink − γ
uβ
+
α + βt
k
+ u βψ k (α + β t ) − wβψ ( p,q ) (α + β t )
µ ′(t ) = wβ In[ p]q − u β
+ uψ k (α + β t ) − wψ ( q,k ) (α + β t )  > 0
as a result of Lemma 2.4. That implies δ is nonincreasing on t ∈ (0, ∞) . Consequently, H is nonincreasing on t ∈ (0, ∞) and for each t ∈ (0,1) we have,

Ink uγ
u
= β  wIn[ p ]q − u
+
+
k
k α + βt

+ uψ k (α + β t ) − wψ ( p,q ) (α + β t )  > 0
as a result of Lemma 2.3. That implies µ is nonincreasing on t ∈ (0, ∞) . Consequently, G is non-
H (0) < H (t ) < H (1)
yielding equation (16).
increasing on t ∈ (0, ∞) and for each t ∈ (0,1) we have,
4. Conclusion
G (0) < G (t ) < G (1)
yielding equation (14).
Theorem 3.4. Define a function H for t ∈ (0, ∞) ,
q ∈ (0,1) and k > 0 by
H (t ) =
(α + β t )u e
u βγ t
k Γ
uβ t
wβ t
k k (1 − q ) k Γ
k (α
+ β t )u
( q , k ) (α
+ β t)
(15)
If we fix u= w= β= 1 in inequalities (10), (12), (14)
and (16), then we respectively obtain the inequalities (5),
(6), (7) and (8) as special cases. By this, the previous
results [7] have been generalized.
Competing Interests
w
The authors have no competing interests.
where u , w , α , β are positive real numbers. Then, H
is increasing on t ∈ (0, ∞) and the inequalities:
α ue
(α
<
−
u βγ t
k Γ
u
k (α )
uβ t
wβ t
−
−
+ β t )u k k (1 − q ) k Γ
[1]
( q , k ) (α )
w
Γ k (α + β t )u
(16)
Γ( q,k ) (α + β t ) w
(α + β )u e
<
(α
References
u βγ (1−t )
k
Γ
u
k (α + β )
u β (1−t )
wβ (1−t )
+ β t )u k k (1 − q ) k Γ( q,k ) (α
+ β )w
are valid for each t ∈ (0,1) .
Proof. Let δ (t ) = InH (t ) for every t ∈ (0, ∞) . Then
(α + β t )u e
δ (t ) = In
k
u βγ t
k Γ
uβ t
wβ t
k (1 − q ) k Γ
k (α
+ β t )u
( q , k ) (α
+ β t )w
u βγ t u β t
wβ t
−
Ink −
In(1 − q)
k
k
k
+ uInΓ k (α + β t ) − wInΓ( q,k ) (α + β t ).
= uIn(α + β t ) +
Then,
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