SOME GENERALIZED INEQUALITIES INVOLVING
THE q-GAMMA FUNCTION
Inequalities Involving the
q-Gamma Function
J. K. PRAJAPAT
S. KANT
Department of Mathematics
Central University of Rajasthan
16, Nav Durga Colony, Opposite Hotel Clarks Amer,
J. L. N. Marg, Jaipur-302017, Rajasthan, India
EMail: jkp_0007@rediffmail.com
Department of Mathematics
Government Dungar College
Bikaner-334001,
Rajasthan, India.
EMail: drskant.2007@yahoo.com
J.K. Prajapat and S. Kant
vol. 10, iss. 4, art. 120, 2009
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14 May, 2009
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Communicated by:
J. Sándor
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2000 AMS Sub. Class.:
33B15.
Key words:
q-Gamma Function.
Abstract:
In this paper we establish some generalized double inequalities involving the qgamma function.
Received:
06 August, 2008
Accepted:
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Contents
1
Introduction and Preliminary Results
3
2
Main Results
7
Inequalities Involving the
q-Gamma Function
J.K. Prajapat and S. Kant
vol. 10, iss. 4, art. 120, 2009
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1.
Introduction and Preliminary Results
The Euler gamma function Γ(x) is defined for x > 0, by
Z ∞
(1.1)
Γ(x) =
e−t tx−1 dt,
0
and the Psi (or digamma) function is defined by
Inequalities Involving the
q-Gamma Function
0
ψ(x) =
(1.2)
Γ (x)
Γ(x)
(x > 0).
J.K. Prajapat and S. Kant
vol. 10, iss. 4, art. 120, 2009
The q-psi function is defined for 0 < q < 1, by
d
log Γq (x),
dx
where the q-gamma function Γq (x) is defined by (0 < q < 1)
ψq (x) =
(1.3)
1−x
Γq (x) = (1 − q)
(1.4)
∞
Y
1 − qi
.
1 − q x+i
i=1
Many properties of the q-gamma function were derived by Askey [2]. The explicit
form of the q-psi function ψq (x) is
ψq (x) = − log(1 − q) + log q
(1.5)
∞
X
i=0
lim Γq (x) = Γ(x)
and
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x+i
q
.
1 − q x+i
In particular
q→1−
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lim ψq (x) = ψ(x).
q→1−
For the gamma function Alsina and Thomas [1] proved the following double
inequality:
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Theorem 1.1. For all x ∈ [0, 1], and all nonnegative integers n, the following double
inequality holds true
1
[Γ(1 + x)]n
≤
≤ 1.
n!
Γ(1 + nx)
(1.6)
Sándor [4] and Shabani [5] proved the following generalizations of (1.6) given
by Theorem 1.2 and Theorem 1.3 respectively.
Theorem 1.2. For all a ≥ 1 and all x ∈ [0, 1], one has
Inequalities Involving the
q-Gamma Function
J.K. Prajapat and S. Kant
vol. 10, iss. 4, art. 120, 2009
(1.7)
1
[Γ(1 + x)]a
≤
≤ 1.
Γ(1 + a)
Γ(1 + ax)
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Theorem 1.3. Let a ≥ b > 0, c, d be positive real numbers such that bc ≥ ad > 0
and ψ(b + ax) > 0, where x ∈ [0, 1]. Then the following double inequality holds:
(1.8)
[Γ(a)]c
[Γ(a + bx)]c
≤
≤ [Γ(a + b)]c−d .
[Γ(b)]d
[Γ(b + ax)]d
Recently, Mansour [3] extended above gamma function inequalities to the case
of Γq (x), given by Theorem 1.4, below:
Theorem 1.4. Let x ∈ [0, 1] and q ∈ (0, 1). If a ≥ b > 0, c, d are positive real
numbers with bc ≥ ad > 0 and ψq (b + ax) > 0, then
c
(1.9)
c
[Γq (a)]
[Γq (a + bx)]
≤
≤ [Γq (a + b)]c−d .
d
d
[Γb (b)]
[Γq (b + ax)]
In our investigation we shall require the following lemmas:
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Lemma 1.5. Let q ∈ (0, 1), α > 0 and a, b be any two positive real numbers such
that a ≥ b. Then
(1.10)
ψq (aα + bx) ≥ ψq (bα + ax)
x ∈ [0, α],
and
(1.11)
ψq (aα + bx) ≤ ψq (bα + ax)
x ∈ [α, ∞).
Proof. By using (1.5), we have
Inequalities Involving the
q-Gamma Function
J.K. Prajapat and S. Kant
∞ X
q bα+ax+i
q aα+bx+i
ψq (aα + bx) − ψq (bα + ax) = log q
−
aα+bx+i
1
−
q
1 − q bα+ax+i
i=0
∞
X
q i q aα+bx − q bα+ax
= log q
(1 − q aα+bx+i )(1 − q bα+ax+i )
i=0
∞
X
q b(x+α)+i q (a−b)α − q (a−b)x
.
= log q
aα+bx+i )(1 − q bα+ax+i )
(1
−
q
i=0
vol. 10, iss. 4, art. 120, 2009
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Since for 0 < q < 1, we have log q < 0. In addition, for a ≥ b, x ∈ [0, α], we get
(1 − q aα+bx+i ) > 0, (1 − q bα+ax+i ) > 0 and q (a−b)α ≤ q (a−b)x . Hence
ψq (aα + bx) ≥ ψq (bα + ax)
x ∈ [0, α].
Furthermore, for a ≥ b and x ∈ [α, ∞), we have (1 − q aα+bx+i ) > 0, (1 −
q bα+ax+i ) > 0 and q (a−b)α ≥ q (a−b)x . Hence
ψq (aα + bx) ≤ ψq (bα + ax)
which completes the proof.
x ∈ [α, ∞).
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Lemma 1.6. Let x ∈ [0, α], α > 0 and q ∈ (0, 1). If a, b, c, d are positive real
numbers such that a ≥ b and [bc ≥ ad, ψq (bα+ax) > 0] or [bc ≤ ad, ψq (aα+bx) <
0], we have
(1.12)
bcψq (aα + bx) − adψq (bα + ax) ≥ 0.
Proof. Since bc ≥ ad and ψq (bα + ax) > 0, then using (1.10), we obtain
adψq (bα + ax) ≤ bcψq (bα + ax)
≤ bcψq (aα + bx).
Inequalities Involving the
q-Gamma Function
J.K. Prajapat and S. Kant
vol. 10, iss. 4, art. 120, 2009
Similarly, when bc ≤ ad and ψq (aα + bx) < 0, we have
bcψq (aα + bx) ≥ adψq (aα + bx) ≥ adψq (bα + ax).
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This proves Lemma 1.6.
Similarly, using (1.11) and a similar proof to that above, we have the following
lemma:
Lemma 1.7. Let q ∈ (0, 1) and x ∈ [α, ∞), α > 0. If a, b, c, d are positive real
numbers such that a ≥ b and [bc ≥ ad, ψq (bα+ax) < 0] or [bc ≤ ad, ψq (aα+bx) <
0], we have
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(1.13)
bcψq (aα + bx) − adψq (bα + ax) ≤ 0.
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2.
Main Results
In this section we will establish some generalized double inequalities involving the
q- gamma function.
Theorem 2.1. For all q ∈ (0, 1), x ∈ [0, α], α > 0 and positive real numbers
a, b, c, d such that a ≥ b and [bc ≥ ad, ψq (bα + ax) > 0] or [bc ≤ ad, ψq (aα + bx) <
0], we have
(2.1)
[Γq (aα)]c
[Γq (aα + bx)]c
≤
≤ [Γq {(a + b)α}]c−d .
[Γq (bα)]d
[Γq (bα + ax)]d
Proof. Let
Inequalities Involving the
q-Gamma Function
J.K. Prajapat and S. Kant
vol. 10, iss. 4, art. 120, 2009
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c
(2.2)
f (x) =
[Γq (aα + bx)]
,
[Γq (bα + ax)]d
and assume that g(x) is a function defined by g(x) = log f (x). Then
g(x) = c log Γq (aα + bx) − d log Γq (bα + ax),
so
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Γ0q (aα + bx)
Γ0q (bα + ax)
0
g (x) = bc
− ad
Γq (aα + bx)
Γq (bα + ax)
= bcψq (aα + bx) − adψq (bα + ax).
Thus using Lemma 1.6, we have g 0 (x) ≥ 0. This means that g(x) is an increasing
function in [0, α], which implies that the function f (x) is also an increasing function
in [0, α], so that
f (0) ≤ f (x) ≤ f (α),
x ∈ [0, α],
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and this is equivalent to
[Γq (aα)]c
[Γq (aα + bx)]c
≤
≤ [Γq {(a + b)α}]c−d .
[Γq (bα)]d
[Γq (bα + ax)]d
This completes the proof of Theorem 2.1.
Theorem 2.2. For all q ∈ (0, 1), x ∈ [α, ∞), α > 0 and positive real numbers
a, b, c, d such that a ≥ b and [bc ≥ ad, ψq (bα + ax) < 0] or [bc ≤ ad, ψq (aα + bx) >
0], we have
(2.3)
[Γq (aα + bx)]c
≤ [Γq (a + b)α]c−d
[Γq (bα + ax)]d
J.K. Prajapat and S. Kant
vol. 10, iss. 4, art. 120, 2009
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and
(2.4)
Inequalities Involving the
q-Gamma Function
[Γq (aα + by)]c
[Γq (aα + bx)]c
≤
,
[Γq (bα + ax)]d
[Γq (bα + ay)]d
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α < y < x.
Proof. Applying Lemma 1.7 and an argument similiar to that of Theorem 2.1, we
see that the function f (x) defined by (2.2) is a decreasing function. Therefore we
have
f (x) ≤ f (α),
x ∈ [α, ∞),
which gives the desired result.
Remark 1.
(i) Taking α = 1, Theorem 2.1 and Theorem 2.2 yield the results obtained by
Mansour [3].
(ii) Taking α = 1 and q → 1− , Theorem 2.1 and Theorem 2.2 yield the results
obtained by Shabani [5].
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References
[1] C. ALSINA AND M.S. THOMAS, A geometrical proof of a new inequality for
the gamma function, J. Inequal. Pure & Appl. Math., 6(2) (2005), Art. 48. [ONLINE: http://jipam.vu.edu.au/article.php?sid=517].
[2] R. ASKEY, The q-gamma and q-beta function, Applicable Anal., 8(2) (1978/79),
125–141.
[3] T. MANSOUR, Some inequalities for q-gamma function, J. Inequal. Pure &
Appl. Math., 9(1) (2008), Art. 18. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=954].
[4] J. SÁNDOR, A note on certain inequalities for the gamma function, J. Inequal. Pure Appl. Math., 6(3) (2005), Art. 61. [ONLINE: http://jipam.
vu.edu.au/article.php?sid=534].
[5] A.S. SHABANI, Some inequalities for the gamma function, J. Inequal. Pure
Appl. Math., 8(2) (2007), Art. 49. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=852].
Inequalities Involving the
q-Gamma Function
J.K. Prajapat and S. Kant
vol. 10, iss. 4, art. 120, 2009
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