Volume 9 (2008), Issue 2, Article 48, 6 pp.
AN INEQUALITY ABOUT 3 φ2 AND ITS APPLICATIONS
MINGJIN WANG AND HONGSHUN RUAN
D EPARTMENT OF I NFORMATION S CIENCE
J IANGSU P OLYTECHNIC U NIVERSITY
C HANGZHOU C ITY 213164
J IANGSU P ROVINCE , P.R. C HINA .
wmj@jpu.edu.cn
rhs@em.jpu.edu.cn
Received 17 August, 2007; accepted 18 May, 2008
Communicated by S.S. Dragomir
A BSTRACT. In this paper, we use the terminating case of the q-binomial formula, the q-ChuVandermonde formula and the Grüss inequality to drive an inequality about 3 φ2 . As applications
of the inequality, we discuss the convergence of some q-series involving 3 φ2 .
Key words and phrases: Basic hypergeometric function 3 φ2 ; q-binomial theorem; q-Chu-Vandermonde formula; Grüss inequality.
2000 Mathematics Subject Classification. Primary 26D15; Secondary 33D15.
1. S TATEMENT OF M AIN R ESULTS
q-series, which are also called basic hypergeometric series, play a very important role in
many fields, such as affine root systems, Lie algebras and groups, number theory, orthogonal
polynomials and physics, etc. Inequalities techniques provide useful tools in the study of special functions (see [1, 6, 7, 8, 9, 10]). For example, Ito used inequalities techniques to give a
sufficient condition for convergence of a special q-series called the Jackson integral [6]. In this
paper, we derive the following new inequality about q-series involving 3 φ2 .
Theorem 1.1. Let a1 , a2 , b1 , b2 be some real numbers such that bi < 1 for i = 1, 2. Then for all
positive integers n, we have:
−n
b
/a
,
b
/a
,
q
(a
,
a
;
q)
1
2
n
1
1
2
2
n
3 φ2
≤ λµ(−1; q)n ,
(1.1)
; q, −a1 a2 q −
b1 , b2
(−1, b1 , b2 ; q)n where
|a1 − b1 |
|a2 − b2 |
n
n
λ = max{1, M }, M = max |a1 |,
, µ = max{1, N }, N = max |a2 |,
.
1 − b1
1 − b2
Applications of this inequality are also given.
The author would like to express deep appreciation to the referee for the helpful suggestions. In particular, the author thanks the referee for
helping to improve the presentation of the paper.
271-07
2
M INGJIN WANG AND H ONGSHUN RUAN
2. N OTATIONS AND K NOWN R ESULTS
We recall some definitions, notations and known results which will be used in the proof.
Throughout this paper, it is supposed that 0 < q < 1. The q-shifted factorials are defined as
(2.1)
(a; q)0 = 1,
(a; q)n =
n−1
Y
k
(1 − aq ),
(a; q)∞ =
k=0
∞
Y
(1 − aq k ).
k=0
We also adopt the following compact notation for multiple q-shifted factorials:
(2.2)
(a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n · · · (am ; q)n ,
where n is an integer or ∞.
The q-binomial theorem (see [2, 3, 4]) is given by
(2.3)
∞
X
(a; q)k z k
k=0
(q; q)k
=
(az; q)∞
,
(z; q)∞
|z| < 1.
When a = q −n , where n denotes a nonnegative integer, we have
(2.4)
n
X
(q −n ; q)k z k
k=0
(q; q)k
= (zq −n ; q)n .
Heine introduced the r+1 φr basic hypergeometric series, which is defined by
X
∞
a1 , a2 , . . . , ar+1
(a1 , a2 , . . . , ar+1 ; q)n xn
(2.5)
; q, x =
.
r+1 φr
(q, b1 , b2 , . . . , br ; q)n
b1 , b2 , . . . , br
n=0
The q-Chu-Vandermonde sums (see [2, 3, 4]) are
a, q −n
an (c/a; q)n
(2.6)
φ
;
q,
q
=
2 1
c
(c; q)n
and, reversing the order of summation, we have
a, q −n
(c/a; q)n
n
(2.7)
; q, cq /a =
.
2 φ1
c
(c; q)n
At the end of this section, we recall the Grüss inequality (see [5]):
Z b
Z b
Z b
1
1
1
(2.8)
f
(x)g(x)dx
−
f
(x)dx
g(x)dx
b − a
b
−
a
b
−
a
a
a
a
(M − m)(N − n)
≤
,
4
provided that f, g : [a, b] → R are integrable on [a, b] and m ≤ f (x) ≤ M, n ≤ g(x) ≤ N for
all x ∈ [a, b], where m, M , n, N are given constants.
By simple calculus, one can prove that the discrete version of the Grüss inequality can be
stated as:
if a ≤ λi ≤ A and b ≤ µi ≤ P
B, i = 1, 2, . . . , n, then for all sequences (pi )0≤i≤n of
nonnegative real numbers satisfying ni=1 pi = 1, we have
n
!
!
n
n
X
(A − a)(B − b)
X
X
(2.9)
λ i µi p i −
λi p i ·
µi p i ≤
,
4
i=1
i=1
i=1
where a, A, b, B are some given real constants.
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 48, 6 pp.
http://jipam.vu.edu.au/
A N I NEQUALITY A BOUT 3 φ2
AND ITS
A PPLICATIONS
3
3. P ROOF OF THE T HEOREM
In this section, we use the terminating case of the q-binomial formula, the q-Chu-Vandermonde
formula and the Grüss inequality to prove (1.1). For this purpose, we need the following lemma.
Lemma 3.1. Let a and b be two real numbers such that b < 1, and let 0 ≤ t ≤ 1. Then,
a − bt |a
−
b|
(3.1)
1 − bt ≤ max |a|, 1 − b .
Proof. Let
f (t) =
a − bt
,
1 − bt
0 ≤ t ≤ 1,
then
f 0 (t) =
b(a − 1)
,
(1 − bt)2
0 ≤ t ≤ 1.
So f (t) is a monotonic function with respect to 0 ≤ t ≤ 1. Since f (0) = a and f (1) =
(3.1) holds.
a−b
,
1−b
Now, we are in a position to prove the inequality (1.1).
Proof. Put
(3.2)
pk =
(q −n ; q)k (−q n )k
,
(q; q)k (−1; q)n
k = 0, 1, . . . , n.
It is obvious that pk ≥ 0.
On the other hand, using (2.4), we obtain
n
X
n
X
1
(q −n ; q)k (−q n )k
pk =
= 1.
(−1;
q)
(q;
q)
n
k
k=0
k=0
Let
(3.3)
λk =
(−a1 )k (b1 /a1 ; q)k
,
(b1 ; q)k
µk =
(−a2 )k (b2 /a2 ; q)k
,
(b2 ; q)k
and
(3.4)
where k = 0, 1, . . . , n.
According to the definitions of M, N, λ and µ, it is easy to see that
M k ≤ λ and N k ≤ µ, 0 ≤ k ≤ n.
Using the lemma, one can get for all 0 ≤ k ≤ n,
a1 − b 1 a1 − b 1 q
a1 − b1 q k−1 (3.5)
|λk | = ·
· ··· ·
≤ Mk ≤ λ
1 − b1 1 − b1 q
1 − b1 q k−1 and
(3.6)
a2 − b 2 a2 − b 2 q
a2 − b2 q k−1 |µk | = · ··· ·
≤ N k ≤ µ.
·
1 − b2 1 − b2 q
1 − b2 q k−1 J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 48, 6 pp.
http://jipam.vu.edu.au/
4
M INGJIN WANG AND H ONGSHUN RUAN
Substitution of (3.2), (3.3), (3.4), (3.5) and (3.6) into (2.9), gives
n
X (q −n ; q) (−q n )k (−a )k (b /a ; q) (−a )k (b /a ; q)
1
1
1
k
2
2
2
k
k
(3.7) ·
·
(q; q)k (−1; q)n
(b1 ; q)k
(b2 ; q)k
k=0
−
n
X
(q −n ; q)k (−q n )k (−a1 )k (b1 /a1 ; q)k
·
(q;
q)
(b1 ; q)k
k (−1; q)n
k=0
n
−n
n k
k
X
(q ; q)k (−q ) (−a2 ) (b2 /a2 ; q)k ×
·
≤ λµ.
(q; q)k (−1; q)n
(b2 ; q)k
k=0
Using (2.5) and (2.7), we get
n
X
(q −n ; q)k (−q n )k (−a1 )k (b1 /a1 ; q)k (−a2 )k (b2 /a2 ; q)k
(3.8)
·
·
(q;
q)
(b
(b2 ; q)k
k (−1; q)n
1 ; q)k
k=0
n
X
1
(q −n , b1 /a1 , b2 /a2 ; q)k
=
(−a1 a2 q n )k
(−1; q)n k=0
(q, b1 , b2 ; q)k
1
b1 /a1 , b2 /a2 , q −n
n
=
; q, −a1 a2 q ,
3 φ2
(−1; q)n
b1 , b2
(3.9)
n
X
(a1 ; q)n
(q −n ; q)k (−q n )k (−a1 )k (b1 /a1 ; q)k
·
=
(q; q)k (−1; q)n
(b1 ; q)k
(−1, b1 ; q)n
k=0
and
(3.10)
n
X
(q −n ; q)k (−q n )k (−a2 )k (b2 /a2 ; q)k
(a2 ; q)n
·
=
.
(q;
q)
(b
(−1,
b
k (−1; q)n
2 ; q)k
2 ; q)n
k=0
Substituting (3.8), (3.9) and (3.10) into (3.7), we obtain (1.1).
Taking a2 = 1 in (1.1), we get the following corollary.
Corollary 3.2. We have
(3.11)
−n
2 φ1 b1 /a1 , q ; q, −a1 q n ≤ λ(−1; q)n .
b1
4. S OME A PPLICATIONS OF THE I NEQUALITY
Convergence of q-series is an important problem which is at times very difficult. As applications of the inequality derived in this paper, we obtain some results about the convergence of
the q-series involving 3 φ2 . In this section, we mainly discuss the convergence of the following
q-series:
∞
X
b1 /a1 , b2 /a2 , q −n
n
(4.1)
cn3 φ2
; q, −a1 a2 q .
b1 , b2
n=0
Theorem 4.1. Suppose |ai | ≤ 1 and bi < ai2+1 for i = 1, 2. Let {cn } be a real sequence
satisfying
cn+1 = p < 1.
lim
n→∞ cn Then the series (4.1) is absolutely convergent.
J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 48, 6 pp.
http://jipam.vu.edu.au/
A N I NEQUALITY A BOUT 3 φ2
AND ITS
A PPLICATIONS
5
Proof. It is obvious that bi < 1 for i = 1, 2. Combining the following inequality
−n
(a1 , a2 ; q)n b
/a
,
b
/a
,
q
1
1
2
2
n
(4.2) 3 φ2
; q, −a1 a2 q − (−1, b1 , b2 ; q)n b1 , b2
b1 /a1 , b2 /a2 , q −n
(a1 , a2 ; q)n n
≤ 3 φ2
; q, −a1 a2 q −
(−1, b1 , b2 ; q)n b1 , b2
with (2.1), shows that
−n
(a1 , a2 ; q)n b
/a
,
b
/a
,
q
1
1
2
2
n
+ λµ(−1; q)n .
3 φ2
(4.3)
; q, −a1 a2 q ≤ (−1, b1 , b2 ; q)n b1 , b2
Since
|ai | ≤ 1,
bi <
ai + 1
,
2
i = 1, 2,
|ai | ≤ 1,
|ai − bi |
< 1,
1 − bi
i = 1, 2,
which is equivalent to
then
λ = µ = 1.
(4.4)
Substituting (4.4) into (4.3), we obtain
−n
(a1 , a2 ; q)n b
/a
,
b
/a
,
q
1
1
2
2
n
3 φ2
+ (−1; q)n .
(4.5)
; q, −a1 a2 q ≤ (−1, b1 , b2 ; q)n b 1 , b2
Multiplication of the two sides of (4.5) by |cn | gives
cn (a1 , a2 ; q)n b1 /a1 , b2 /a2 , q −n
n + |cn (−1; q)n | .
(4.6)
; q, −a1 a2 q ≤ cn3 φ2
b 1 , b2
(−1, b1 , b2 ; q)n The ratio test shows that both
∞
X
cn (a1 , a2 ; q)n
(−1, b1 , b2 ; q)n
n=0
and
∞
X
cn (−1; q)n
n=0
are absolutely convergent. From (4.6), we get that (4.1) is absolutely convergent.
Theorem 4.2. Suppose |a1 | > 1 or a1 < 2b1 − 1, b1 < 1, |a2 | ≤ 1 and b2 ≤ a22+1 . Let {cn } be
a real sequence satisfying
cn+1 =p< 1 ,
lim n→∞
cn M
n
o
1 −b1 |
where M = max |a1 |, |a1−b
. Then the series (4.1) is absolutely convergent.
1
Proof. First we point out that a1 < 2b1 − 1 implies
|a1 − b1 |
> 1.
1 − b1
So, under the conditions of the theorem, we know
λ = Mn
and
µ = 1.
Multiplying both sides of (4.3) by |cn |, one gets
cn (a1 , a2 ; q)n b1 /a1 , b2 /a2 , q −n
n + |cn M n (−1; q)n | .
(4.7)
; q, −a1 a2 q ≤ cn3 φ2
b 1 , b2
(−1, b1 , b2 ; q)n J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 48, 6 pp.
http://jipam.vu.edu.au/
6
M INGJIN WANG AND H ONGSHUN RUAN
The ratio test shows that both
∞
X
cn (a1 , a2 ; q)n
(−1, b1 , b2 ; q)n
n=0
and
∞
X
cn M n (−1; q)n
n=0
are absolutely convergent. From (4.7), we get that (4.1) is absolutely convergent.
Similarly, we have
Theorem 4.3. Suppose |ai | > 1 or ai < 2bi − 1, bi < 1 with i = 1, 2. Let {cn } be a real
sequence satisfying
cn+1 =p< 1 ,
lim n→∞ cn MN
o
n
o
n
|a2 −b2 |
|a1 −b1 |
where M = max |a1 |, 1−b1 and N = max |a2 |, 1−b2 . Then the series (4.1) is absolutely convergent.
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J. Inequal. Pure and Appl. Math., 9(2) (2008), Art. 48, 6 pp.
http://jipam.vu.edu.au/