NEW INEQUALITIES FOR SOME SPECIAL AND
q-SPECIAL FUNCTIONS
MOUNA SELLAMI
KAMEL BRAHIM
Institut Préparatoire aux Études d’Ingénieur de El Manar
Tunis, Tunisia
EMail: mouna.sellami@ipeimt.rnu.tn
Institut Préparatoire aux Études d’Ingénieur de Tunis
Tunis, Tunisia
EMail: kamel.brahim@ipeit.rnu.tn
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
NÉJI BETTAIBI
Institut Préparatoire aux Études d’Ingénieur de Mounastir,
5000 Mounastir, Tunisia.
EMail: Neji.Bettaibi@ipein.rnu.tn
Contents
JJ
II
I
Received:
14 February, 2007
J
Accepted:
25 May, 2007
Page 1 of 16
Communicated by:
S.S. Dragomir
Go Back
2000 AMS Sub. Class.:
33B15, 33D05.
Key words:
Gamma function, Beta function, q-Gamma function, q-Beta function, q-Zeta
function.
Full Screen
Abstract:
In this paper, we give new inequalities involving some special (resp. q-special)
functions, using their integral (resp. q-integral) representations and a technique
developed by A. McD. Mercer in [11]. These inequalities generalize those given
in [1], [2], [7] and [11].
Close
Contents
1
Introduction and Preliminaries
3
2
The Gamma Function
6
3
The q-Gamma Function
8
4
The q-Beta function
11
5
The q- Zeta Function
13
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
JJ
II
J
I
Page 2 of 16
Go Back
Full Screen
Close
1.
Introduction and Preliminaries
In [1], Alsina and M. S. Tomas studied a very interesting inequality involving the
Gamma function and they proved the following double inequality
1
Γ(1 + x)n
≤
≤ 1,
n!
Γ(1 + nx)
(1.1)
x ∈ [0, 1], n ∈ N,
by using geometric method.
In view of the interest in this type of inequalities, many authors extended this
result to more general cases either for the classical Gamma function or the basic
one, by using geometric or analytic approaches (see [2], [7], [12]).
In [11], A. McD. Mercer, developed a very interesting technique which was the
source of some inequalities involving the Gamma, Beta and Zeta functions.
He considered a positive linear functional L defined on a subspace C ∗ (I) of C(I)
(the space of continuous functions on I), where I is the interval (0, a) with a > 0 or
equal to +∞, and he proved the following result:
Theorem 1.1. For f, g in C ∗ (I) such that f (x) → 0, g(x) → 0 as x → 0+ and
strictly increasing, put
L(f )
φ=g
L(g)
f
g
is
and let F be defined on the ranges of f and g such that the compositions F (f ) and
F (g) each belong to C ∗ (I).
a) If F is convex then
(1.2)
L [F (f )] ≥ L [F (φ)] .
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
JJ
II
J
I
Page 3 of 16
Go Back
Full Screen
Close
b) If F is concave then
L [F (f )] ≤ L [F (φ)] .
(1.3)
In this paper, using the previous theorem, we obtain some generalizations of inequalities involving some special and q-special functions.
Note that for α ∈ R, the function
F (t) = tα
is convex if α < 0 or α > 1 and concave if 0 < α < 1.
So, for f and g satisfying the conditions of the previous theorem, we have:
L(f α ) > L(φα ) if α < 0 or α > 1 and L(f α ) < L(φα ) if
0 < α < 1.
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
Substituting for φ this reads:
[L(g)]α
[L(f )]α
>
(resp.
<)
,
L(g α )
L(f α )
JJ
II
J
I
if α < 0 or α > 1 (resp. 0 < α < 1). In particular, if we take f (x) = xβ and
g(x) = xδ with β > δ > 0, we obtain the following useful inequality:
Page 4 of 16
[L(xβ )]α
[L(xδ )]α
≷
,
L(xαδ )
L(xαβ )
Full Screen
(1.4)
where, we follow the notations of [11], and ≷ correspond to the case (α < 0 or
α > 1) and ( 0 < α < 1) respectively.
Throughout this paper, we will fix q ∈]0, 1[ and we will follow the terminology
and notation of the book by G. Gasper and M. Rahman [4]. We denote, in particular,
Go Back
Close
for a ∈ C
1 − qa
[a]q =
,
1−q
(a; q)n =
n−1
Y
(1 − aq k ), n = 1, 2, . . . , ∞.
k=0
The q-Jackson integrals from 0 to a and from 0 to ∞ are defined by (see [5])
Z a
∞
X
(1.5)
f (x)dq x = (1 − q)a
f (aq n )q n ,
0
n=0
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Z
∞
f (x)dq x = (1 − q)
(1.6)
0
∞
X
f (q n )q n ,
Title Page
n=−∞
Contents
provided the sums converge absolutely.
The q-Jackson integral in a generic interval [a, b] is given by (see [5])
Z b
Z b
Z a
(1.7)
f (x)dq x =
f (x)dq x −
f (x)dq x.
a
0
0
JJ
II
J
I
Page 5 of 16
Go Back
Full Screen
Close
2.
The Gamma Function
Theorem 2.1. Let f be the function defined by
(2n)
α
Γ (1 + x)
(2.1)
f (x) = (2n)
Γ (1 + αx)
then for all 0 < α < 1 (resp. α > 1) f is increasing (resp. decreasing) on (0, ∞).
Proof. First, we recall that the Gamma function is infinitely differentiable on ]0, +∞[
and we have
Z ∞
(n)
∀x ∈]0, +∞[, ∀n ∈ N, Γ (x) =
tx−1 [Log(t)]n e−t dt.
0
Now, we consider the subspace C ∗ (I) obtained from C(I) by requiring its members
to satisfy:
(i) w(x) = O(xθ ) (for any θ > −1) as x → 0,
(ii) w(x) = O(xϕ ) (for any finite ϕ) as x → +∞.
For w ∈ C ∗ (I), we define
Z ∞
(2.2)
L(w) =
w(x)(Log(x))2n e−x dx.
0
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
JJ
II
J
I
Page 6 of 16
Go Back
Full Screen
∗
The linear functional L is well-defined on C (I) and it is positive.
Then, by applying the inequality (1.4), we obtain for β > δ > 0,
(2n)
α
(2n)
α
Γ (1 + δ)
Γ (1 + β)
(2.3)
≷ (2n)
.
Γ(2n) (1 + αδ)
Γ (1 + αβ)
This completes the proof.
Close
In particular, we have the following result, which generalizes inequality (4.1) of
[11].
Corollary 2.2. For all x ∈ [0, 1] we have:
(2n) α
(2n)
α
α−1
Γ (2)
Γ (1 + x)
(2.4)
≤ (2n)
≤ Γ(2n) (1)
(2n)
Γ (1 + α)
Γ (1 + αx)
if
α≥1
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
and
(2.5)
(2n)
α
(2n) α
(2n) α−1
Γ (1 + x)
Γ (2)
Γ (1)
≤ (2n)
≤ (2n)
Γ (1 + αx)
Γ (1 + α)
if
0 < α ≤ 1.
vol. 8, iss. 2, art. 47, 2007
Taking n = 0, one obtains:
Title Page
Corollary 2.3. For all x ∈ [0, 1],
Contents
(2.6)
[Γ(1 + x)]α
1
≤
≤ 1,
Γ(1 + α)
Γ(1 + αx)
if
α ≥ 1,
and
II
J
I
Page 7 of 16
α
(2.7)
JJ
1≤
[Γ(1 + x)]
1
≤
,
Γ(1 + αx)
Γ(1 + α)
if
0 < α ≤ 1.
Go Back
Full Screen
Close
3.
The q-Gamma Function
Jackson [5] defined a q-analogue of the Gamma function by
(3.1)
Γq (x) =
(q; q)∞
(1 − q)1−x ,
(q x ; q)∞
x 6= 0, −1, −2, . . . .
It is well known that it satisfies
(3.2) Γq (x+1) = [x]q Γq (x),
Γq (1) = 1 and
lim Γq (x) = Γ(x), <(x) > 0.
q→1−
It has the following q-integral representation (see [8])
Z 1
1−q
ts−1 Eq−qt dq t,
(3.3)
Γq (s) =
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
0
where
(3.4) Eqz =0 ϕ0 (−; −; q, −(1 − q)z) =
∞
X
q
n(n−1)
2
n=0
(1 − q)n n
z = (−(1 − q)z; q)∞ ,
(q; q)n
is a q-analogue of the exponential function (see [4] and [6]).
In [3], the authors proved that Γq is infinitely differentiable on ]0, +∞[ and we
have
Z 1
1−q
(n)
(3.5)
∀x ∈]0, +∞[, ∀n ∈ N, Γq (x) =
tx−1 [Log(t)]n Eq−qt dt.
0
Now, we are able to state a q-analogue of Theorem 2.1, and give generalizations
of some inequalities studied in [7].
JJ
II
J
I
Page 8 of 16
Go Back
Full Screen
Close
Theorem 3.1. Let f be the function defined by
h
iα
(2n)
Γq (1 + x)
(3.6)
f (x) = (2n)
Γq (1 + αx)
then for all 0 < α < 1 (resp. α > 1) f is increasing (resp. decreasing) on (0, ∞).
1
Proof. We consider I = 0, 1−q
and the subspace C ∗ (I) obtained from C(I) by
requiring its members to satisfy:
(i) w(x) = O(xθ ) (for any θ > −1) as x → 0,
1
(ii) w(x) = O(1) as x → 1−q
.
∗
For w ∈ C (I), we define
Z 1
1−q
(3.7)
L(w) =
w(x)(Log(x))2n Eq−qx dq x.
0
L is well-defined on C ∗ (I) and it is a positive linear functional on C ∗ (I).
From the inequality (1.4) and the relation (3.5), we obtain for β > δ > 0
h
iα
h
iα
(2n)
(2n)
Γq (1 + δ)
Γq (1 + β)
(3.8)
≷ (2n)
,
(2n)
Γq (1 + αδ)
Γq (1 + αβ)
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
JJ
II
J
I
Page 9 of 16
Go Back
Full Screen
which achieves the proof.
Close
In particular, we have the following result.
Corollary 3.2. For all x ∈ [0, 1] we have
h
iα
h
iα
(2n)
(2n)
Γq (2)
Γq (1 + x)
α−1
(3.9)
≤ (2n)
≤ Γ(2n)
(1)
,
q
(2n)
Γq (1 + α)
Γq (1 + αx)
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
if
α≥1
and
h
(3.10)
(2n) α−1
≤
Γq (1)
(2n)
Γq
(2n)
Γq
iα
(1 + x)
h
≤
(1 + αx)
(2n)
Γq
(2n)
Γq
(2)
iα
,
(1 + α)
if
0 < α ≤ 1.
Corollary 3.3. For all x ∈ [0, 1],
(3.11)
[Γq (1 + x)]α
1
≤
≤ 1,
Γq (1 + α)
Γq (1 + αx)
if
α ≥ 1,
vol. 8, iss. 2, art. 47, 2007
and
(3.12)
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
1≤
[Γq (1 + x)]α
1
≤
,
Γq (1 + αx)
Γq (1 + α)
if
0 < α ≤ 1.
Title Page
Contents
Proof. By taking n = 0 in Corollary 3.2 we obtain the inequalities (3.11) and (3.12).
JJ
II
J
I
Page 10 of 16
Go Back
Full Screen
Close
4.
The q-Beta function
The q-Beta function is defined by (see [4], [8])
Z 1
(xq; q)∞
dq x,
(4.1)
Bq (t, s) =
xt−1
(xq s ; q)∞
0
<(s) > 0, <(t) > 0
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
and we have
(4.2)
Bq (t, s) =
Γq (t)Γq (s)
.
Γq (t + s)
vol. 8, iss. 2, art. 47, 2007
Since Bq is a q-analogue of the classical Beta function, we can see the following
results as generalizations of those given in [11].
Title Page
Theorem 4.1. For s > 0, let f be the function defined by
Contents
(4.3)
f (x) =
[Bq (1 + x, s)]α
.
Bq (1 + αx, s)
If 0 < α < 1, f is increasing on [0, +∞[.
If α > 1 f is decreasing on [0, +∞[.
JJ
II
J
I
Page 11 of 16
Go Back
∗
Proof. We consider the interval I = (0, 1) and the subspace C (I) obtained from
C(I) by requiring its members to satisfy:
(i) w(x) = O(xθ ) (for any θ > −1) as x → 0,
(ii) w(x) = O(1) as x → 1.
For s > 0, we put for w ∈ C ∗ (I),
Z 1
(xq; q)∞
(4.4)
L(w) =
w(x)
dq x.
(xq s ; q)∞
0
Full Screen
Close
It is easy to see that L is well-defined on C ∗ (I) and it is a positive linear functional
on C ∗ (I).
Then, from the inequality (1.4), we obtain for β > δ > 0
[Bq (1 + δ, s)]α
[Bq (1 + β, s)]α
≷
.
Bq (1 + αδ, s)
Bq (1 + αβ, s)
(4.5)
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
This achieves the proof.
Corollary 4.2. For all x ∈ [0, 1], s > 0
(4.6)
[α + s]q
[Bq (1 + x, s)]α
1
≤
≤ α−1 ,
α
α
[α]q [s]q [s + 1]q Bq (α, s)
Bq (1 + αx, s)
[s]q
vol. 8, iss. 2, art. 47, 2007
if
α ≥ 1.
Title Page
Proof. It is a consequence of the previous theorem and the relations:
Bq (1, s) =
1
,
[s]q
Bq (2, s) =
and
Bq (1 + α, s) =
1
[s]q [s + 1]q
[α]q
Bq (α, s).
[α + s]q
Contents
JJ
II
J
I
Page 12 of 16
Go Back
Full Screen
Close
5.
The q- Zeta Function
For x > 0, we put
Log(x)
α(x) =
−E
Log(q)
and
{x}q =
Log(x)
Log(q)
[x]q
q
,
x+α([x]q )
where E Log(x)
is the integer part of Log(x)
.
Log(q)
Log(q)
In [3], the authors defined the q-Zeta function as follows
ζq (s) =
(5.1)
∞
X
n=1
∞
X q (n+α([n]q ))s
1
=
.
s
{n}sq
[n]
q
n=1
They proved that it is a q-analogue of the classical Riemann Zeta function and in the
additional assumption Log(1−q)
∈ Z, we have for all s ∈ C such that <(s) > 1,
Log(q)
Z ∞
1
ts−1 Zq (t)dq t,
ζq (s) =
eq (s) 0
Γ
where for all t > 0,
Zq (t) =
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
JJ
II
J
I
Page 13 of 16
Go Back
Full Screen
∞
X
n=1
qt
e−{n}
q
and
t
1−t
eq (t) = Γq (t)(−q , −q ; q)∞ .
Γ
(−q, −1; q)∞
Now, we consider the subspace C ∗ (I) obtained from C(I) by requiring its members
to satisfy:
(i) w(x) = O(xθ ) (for any θ > −1) as x → 0,
Close
(ii) w(x) = O(xϕ ) (for any finite ϕ) as x → +∞.
For w ∈ C ∗ (I), we define
Z ∞
(5.2)
L(w) =
w(x)Zq (x)dq x.
0
L is a positive linear functional on C ∗ (I). So, by application of the inequality (1.4),
we obtain for all β > δ > 0,
h
iα
h
iα
e
e
Γq (1 + β)ζq (1 + β)
Γq (1 + δ)ζq (1 + δ)
≷
.
eq (1 + αδ)ζq (1 + αδ)
eq (1 + αβ)ζq (1 + αβ)
Γ
Γ
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
JJ
II
J
I
Page 14 of 16
Go Back
Full Screen
Close
References
[1] C. ALSINA AND M.S. TOMAS, A geometrical proof of a new inequality for
the Gamma function, J. Ineq. Pure. App. Math., 6(2) (2005), Art. 48. [ONLINE:
http://jipam.vu.edu.au/article.php?sid=518].
[2] L. BOUGOFFA, Some Inequalities involving the Gamma function, J. Ineq.
Pure. App. Math., 7(5) (2006), Art. 179. [ONLINE: http://jipam.vu.
edu.au/article.php?sid=796].
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
[3] A. FITOUHI, N. BETTAIBI AND K. BRAHIM, The Mellin transform in quantum calculus, Constructive Approximation, 23(3) (2006), 305–323.
vol. 8, iss. 2, art. 47, 2007
[4] G. GASPER AND M. RAHMAN, Basic Hypergeometric Series, Encyclopedia of Mathematics and its application, Vol. 35, Cambridge Univ. Press, Cambridge, UK, 1990.
Title Page
[5] F.H. JACKSON, On a q-definite integrals, Quarterly Journal of Pure and Applied Mathematics, 41 (1910), 193–203.
[6] V.G. KAC AND P. CHEUNG, Quantum Calculus, Universitext, SpringerVerlag, New York, (2002).
[7] TAEKYUN KIM AND C. ADIGA, On the q-analogue of gamma functions and
related inequalities, J. Inequal. Pure Appl. Math., 6(4) (2005), Art. 118. [ONLINE: http://jipam.vu.edu.au/article.php?sid=592].
[8] T.H. KOORNWINDER, q-Special Functions, a Tutorial, in Deformation Theory and Quantum Groups with Applications to Mathematical Physics, M. Gerstenhaber and J. Stasheff (Eds.), Contemp. Math., 134 (1992), Amer. Math.
Soc.
Contents
JJ
II
J
I
Page 15 of 16
Go Back
Full Screen
Close
[9] T.H. KOORNWINDER, Special functions and q-commuting variables, in Special Functions, q-Series and Related Topics, M.E.H. Ismail, D.R. Masson and
M. Rahman (Eds.), Fields Institute Communications 14, American Mathematical Society, (1997), pp. 131–166; arXiv:q-alg/9608008.
[10] T.H. KOORNWINDER AND R.F. SWARTTOUW, On q-analogues of the
Fourier and Hankel transforms, Trans. Amer. Math. Soc., 333 (1992), 445–461.
[11] A. McD. MERCER, Some new inequalites for the Gamma, Beta and Zeta
functions, J. Ineq. Pure. App. Math., 7(1) (2006), Art. 29. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=636].
[12] J. SÁNDOR, A note on certain inequalities for the Gamma function, J. Ineq.
Pure. App. Math., 6(3) (2005), Art. 61. [ONLINE: http://jipam.vu.
edu.au/article.php?sid=534].
Inequalities for Special
and q-Special Functions
Mouna Sellami, Kamel Brahim
and Néji Bettaibi
vol. 8, iss. 2, art. 47, 2007
Title Page
Contents
JJ
II
J
I
Page 16 of 16
Go Back
Full Screen
Close