AN INEQUALITY ABOUT 3φ2 AND ITS
APPLICATIONS
MINGJIN WANG AND HONGSHUN RUAN
Department of Information Science
Jiangsu Polytechnic University
Changzhou City 213164
Jiangsu Province, P.R. China.
EMail: wmj@jpu.edu.cn
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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rhs@em.jpu.edu.cn
Received:
17 August, 2007
Accepted:
18 May, 2008
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
Primary 26D15; Secondary 33D15.
Key words:
Basic hypergeometric function 3 φ2 ; q-binomial theorem; q-Chu-Vandermonde
formula; Grüss inequality.
Abstract:
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 .
Acknowledgements:
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.
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Contents
1
Statement of Main Results
3
2
Notations and Known Results
4
3
Proof of the Theorem
6
4
Some Applications of the Inequality
10
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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1.
Statement of Main Results
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 .
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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
1
2
2
1
2
n
≤ λµ(−1; q)n ,
(1.1) 3 φ2
; q, −a1 a2 q n −
b1 , b2
(−1, b1 , b2 ; q)n where
n
λ = max{1, M },
µ = max{1, N n },
|a1 − b1 |
M = max |a1 |,
,
1 − b1
|a2 − b2 |
N = max |a2 |,
.
1 − b2
Applications of this inequality are also given.
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2.
Notations and Known Results
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
(1 − aq k ),
(a; q)∞ =
k=0
∞
Y
(1 − aq k ).
k=0
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
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 ,
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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)∞
Contents
|z| < 1.
When a = q −n , where n denotes a nonnegative integer, we have
(2.4)
vol. 9, iss. 2, art. 48, 2008
n
X
(q −n ; q)k z k
k=0
(q; q)k
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= (zq
−n
; q)n .
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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
b1 , b2 , . . . , br
(q, b1 , b2 , . . . , br ; q)n
n=0
The q-Chu-Vandermonde sums (see [2, 3, 4]) are
a, q −n
an (c/a; q)n
(2.6)
;
q,
q
=
φ
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 a
b−a a
b−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 ≤ B, i =P
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
µ
p
−
λ
p
·
µ
p
,
i i i
i i
i i ≤
4
i=1
i=1
i=1
where a, A, b, B are some given real constants.
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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3.
Proof of the Theorem
In this section, we use the terminating case of the q-binomial formula, the q-ChuVandermonde 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) =
then
f 0 (t) =
a − bt
,
1 − bt
b(a − 1)
,
(1 − bt)2
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
Title Page
0 ≤ t ≤ 1,
Contents
0 ≤ t ≤ 1.
So f (t) is a monotonic function with respect to 0 ≤ t ≤ 1. Since f (0) = a and
f (1) = a−b
, (3.1) holds.
1−b
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Now, we are in a position to prove the inequality (1.1).
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Proof. Put
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−n
(3.2)
pk =
It is obvious that pk ≥ 0.
n k
(q ; q)k (−q )
,
(q; q)k (−1; q)n
k = 0, 1, . . . , n.
On the other hand, using (2.4), we obtain
n
X
n
X
(q −n ; q)k (−q n )k
1
pk =
= 1.
(−1; q)n k=0
(q; q)k
k=0
Let
(3.3)
λk =
(−a1 )k (b1 /a1 ; q)k
,
(b1 ; q)k
(−a2 )k (b2 /a2 ; q)k
,
µk =
(b2 ; q)k
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,
k−1 a1 − b 1 a1 − b 1 q
a
−
b
q
1
1
≤ Mk ≤ λ
(3.5)
|λk | = ·
· ··· ·
1 − b1 1 − b1 q
1 − b1 q k−1 Title Page
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and
(3.6)
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
and
(3.4)
Inequality About 3 φ2
Mingjin Wang and
k−1 a2 − b 2 a2 − b 2 q
a
−
b
q
2
2
≤ N k ≤ µ.
·
· ··· ·
|µk | = 1 − b2 1 − b2 q
1 − b2 q k−1 Substitution of (3.2), (3.3), (3.4), (3.5) and (3.6) into (2.9), gives
n
X
(q −n ; q)k (−q n )k (−a1 )k (b1 /a1 ; q)k (−a2 )k (b2 /a2 ; q)k
(3.7) ·
·
(q;
q)
(b
(b2 ; q)k
k (−1; q)n
1 ; q)k
k=0
n
X
(q −n ; q)k (−q n )k (−a1 )k (b1 /a1 ; q)k
−
·
(q; q)k (−1; q)n
(b1 ; q)k
k=0
×
n
X
k=0
(q −n ; q)k (−q n )k (−a2 )k (b2 /a2 ; q)k ·
≤ λµ.
(q; q)k (−1; q)n
(b2 ; q)k
Using (2.5) and (2.7), we get
(3.8)
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
Title Page
n
X
(q −n ; q)k (−q n )k (−a1 )k (b1 /a1 ; q)k (−a2 )k (b2 /a2 ; q)k
·
·
(q; q)k (−1; q)n
(b1 ; q)k
(b2 ; q)k
k=0
n
X
(q −n , b1 /a1 , b2 /a2 ; q)k
1
=
(−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
b1 , b2
(−1; q)n
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(3.9)
n
X
k=0
and
(q −n ; q)k (−q n )k (−a1 )k (b1 /a1 ; q)k
(a1 ; q)n
·
=
(q; q)k (−1; q)n
(b1 ; q)k
(−1, b1 ; q)n
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(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
−n
b
/a
,
q
1
1
n
2 φ1
(3.11)
; q, −a1 q ≤ λ(−1; q)n .
b1
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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4.
Some Applications of the Inequality
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 .
b
,
b
1
2
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.
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 b 1 , b2
−n
b
/a
,
b
/a
,
q
(a
,
a
;
q)
1
1
2
2
1
2
n
n
≤ 3 φ2
; q, −a1 a2 q −
b 1 , b2
(−1, b1 , b2 ; q)n with (2.1), shows that
−n
(a1 , a2 ; q)n b
/a
,
b
/a
,
q
1
1
2
2
n
+ λµ(−1; q)n .
(4.3) 3 φ2
; q, −a1 a2 q ≤ b1 , b2
(−1, b1 , b2 ; q)n Since
|ai | ≤ 1,
bi <
ai + 1
,
2
i = 1, 2,
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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which is equivalent to
|ai | ≤ 1,
|ai − bi |
< 1,
1 − bi
i = 1, 2,
then
(4.4)
λ = µ = 1.
Substituting (4.4) into (4.3), we obtain
(a1 , a2 ; q)n b1 /a1 , b2 /a2 , q −n
n + (−1; q)n .
(4.5)
; q, −a1 a2 q ≤ 3 φ2
b1 , b2
(−1, b1 , b2 ; q)n Multiplication of the two sides of (4.5) by |cn | gives
−n
b
/a
,
b
/a
,
q
1
1
2
2
n
cn3 φ2
(4.6)
;
q,
−a
a
q
1
2
b1 , b2
cn (a1 , a2 ; q)n + |cn (−1; q)n | .
≤
(−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
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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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
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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
−n
b
/a
,
b
/a
,
q
1
1
2
2
n
cn3 φ2
(4.7)
;
q,
−a
a
q
1
2
b1 , b2
cn (a1 , a2 ; q)n + |cn M n (−1; q)n | .
≤ (−1, b1 , b2 ; q)n The ratio test shows that both
∞
X
cn (a1 , a2 ; q)n
n=0
(−1, b1 , b2 ; q)n
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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and
∞
X
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n
cn M (−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
n
o
n
o
|a2 −b2 |
1 −b1 |
where M = max |a1 |, |a1−b
and
N
=
max
|a
|,
. Then the series (4.1)
2
1−b2
1
is absolutely convergent.
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References
[1] G.D. ANDERSON, R.W. BARNARD, K.C. VAMANAMURTHY AND M.
VUORINEN, Inequalities for zero-balanced hypergeometric functions, Transactions of the American Mathematical Society, 347(5), 1995.
[2] G.E. ANDREWS, The Theory of Partitions, Encyclopedia of Mathematics and
Applications; Volume 2. Addison-Wesley Publishers, 1976.
[3] W.N. BAILEY, Generalized hypergeometric series, Cambridge Math. Tract
No.32, Cambridge University Press, London and New York.1960.
[4] G. GASPER AND M. RAHMAN, Basic Hypergeometric Series, Cambridge
Univ.Press, Cambridge, MA, 1990.
[5] G. R GRÜSS. Über dasR maximum Rdes absoluten Betrages von
b
b
b
1
1
1
f (x)g(x)dx − ( b−a
f (x)dx)( b−a
g(x)dx), Math.Z., 39 (1935),
b−a a
a
a
215–226.
[6] M. ITO, Convergence and asymptotic behavior of Jackson integrals associated
with irreducible reduced root systems, Journal of Approximation Theory, 124
(2003) 154–180.
[7] MINGJIN WANG, An inequality about q-series, J. Ineq. Pure and Appl.
Math., 7(4) (2006), Art. 136. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=756].
[8] MINGJIN WANG, An Inequality and its q-analogue, J. Ineq. Pure and Appl.
Math., 8(2) (2007), Art. 50. [ONLINE: http://jipam.vu.edu.au/
article.php?sid=853].
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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[9] MINGJIN WANG, An inequality for r+1 φr and its applications, Journal of
Mathematical Inequalities, 1(2007) 339–345.
[10] MINGJIN WANG, Two inequalities for r φr and applications, Journal of Inequalities and Applications, 2008, Article ID 471527.
Inequality About 3 φ2
Mingjin Wang and
Hongshun Ruan
vol. 9, iss. 2, art. 48, 2008
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