Volume 10 (2009), Issue 4, Article 120, 4 pp.
SOME GENERALIZED INEQUALITIES INVOLVING THE q-GAMMA FUNCTION
J. K. PRAJAPAT AND S. KANT
D EPARTMENT OF M ATHEMATICS
C ENTRAL U NIVERSITY OF R AJASTHAN
16, NAV D URGA C OLONY, O PPOSITE H OTEL C LARKS A MER ,
J. L. N. M ARG , JAIPUR -302017, R AJASTHAN , I NDIA
jkp_0007@rediffmail.com
D EPARTMENT OF M ATHEMATICS
G OVERNMENT D UNGAR C OLLEGE
B IKANER -334001, R AJASTHAN , I NDIA .
drskant.2007@yahoo.com
Received 06 August, 2008; accepted 14 May, 2009
Communicated by J. Sándor
A BSTRACT. In this paper we establish some generalized double inequalities involving the qgamma function.
Key words and phrases: q-Gamma Function.
2000 Mathematics Subject Classification. 33B15.
1. I NTRODUCTION AND P RELIMINARY R ESULTS
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
(1.2)
ψ(x) =
Γ0 (x)
Γ(x)
(x > 0).
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)
(1.3)
(1.4)
221-08
ψq (x) =
1−x
Γq (x) = (1 − q)
∞
Y
1 − qi
.
1 − q x+i
i=1
2
J. K. P RAJAPAT AND S. K ANT
Many properties of the q-gamma function were derived by Askey [2]. The explicit form of
the q-psi function ψq (x) is
∞
X
q x+i
(1.5)
ψq (x) = − log(1 − q) + log q
.
1 − q x+i
i=0
In particular
lim Γq (x) = Γ(x)
q→1−
and
lim ψq (x) = ψ(x).
q→1−
For the gamma function Alsina and Thomas [1] proved the following double inequality:
Theorem 1.1. For all x ∈ [0, 1], and all nonnegative integers n, the following double inequality
holds true
[Γ(1 + x)]n
1
(1.6)
≤
≤ 1.
n!
Γ(1 + nx)
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
1
[Γ(1 + x)]a
(1.7)
≤
≤ 1.
Γ(1 + a)
Γ(1 + ax)
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:
[Γ(a + bx)]c
[Γ(a)]c
≤
≤ [Γ(a + b)]c−d .
(1.8)
[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
[Γq (a)]c
[Γq (a + bx)]c
≤
≤ [Γq (a + b)]c−d .
d
d
[Γb (b)]
[Γq (b + ax)]
In our investigation we shall require the following lemmas:
(1.9)
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
∞ X
q aα+bx+i
q bα+ax+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
.
(1 − q aα+bx+i )(1 − q bα+ax+i )
i=0
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 120, 4 pp.
http://jipam.vu.edu.au/
I NEQUALITIES I NVOLVING THE q-G AMMA F UNCTION
3
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
≥ q (a−b)x . Hence
(a−b)α
ψq (aα + bx) ≤ ψq (bα + ax)
x ∈ [α, ∞).
which completes the proof.
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
bcψq (aα + bx) − adψq (bα + ax) ≥ 0.
(1.12)
Proof. Since bc ≥ ad and ψq (bα + ax) > 0, then using (1.10), we obtain
adψq (bα + ax) ≤ bcψq (bα + ax)
≤ bcψq (aα + bx).
Similarly, when bc ≤ ad and ψq (aα + bx) < 0, we have
bcψq (aα + bx) ≥ adψq (aα + bx) ≥ adψq (bα + ax).
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
(1.13)
bcψq (aα + bx) − adψq (bα + ax) ≤ 0.
2. M AIN R ESULTS
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
(2.2)
[Γq (aα + bx)]c
f (x) =
,
[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
Γ0q (aα + bx)
Γ0q (bα + ax)
− ad
Γq (aα + bx)
Γq (bα + ax)
= bcψq (aα + bx) − adψq (bα + ax).
g 0 (x) = bc
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 120, 4 pp.
http://jipam.vu.edu.au/
4
J. K. P RAJAPAT AND S. K ANT
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],
and this is equivalent to
[Γq (aα)]c
[Γq (aα + bx)]c
≤
≤ [Γq {(a + b)α}]c−d .
d
d
[Γq (bα)]
[Γq (bα + ax)]
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
[Γq (aα + bx)]c
≤ [Γq (a + b)α]c−d
[Γq (bα + ax)]d
(2.3)
and
(2.4)
[Γq (aα + bx)]c
[Γq (aα + by)]c
≤
,
[Γq (bα + ax)]d
[Γq (bα + ay)]d
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].
R EFERENCES
[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].
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 120, 4 pp.
http://jipam.vu.edu.au/