An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page
Contents
JJ II
J I
Page1of 14 Go Back Full Screen
Close
AN INEQUALITY AND ITS q-ANALOGUE
MINGJIN WANG
Department of Mathematics East China Normal University, Shanghai, 200062,
People’s Republic of China EMail:wmj@jpu.edu.cn
Received: 16 July, 2006
Accepted: 11 June, 2007
Communicated by: J. Sándor
2000 AMS Sub. Class.: Primary 26D15; Secondary 33D15.
Key words: Gould-Hsu inversions; Carlitz inversions; Grüss inequality;q-series.
Abstract: In this paper, we establish a new inequality and itsq-analogue by means of the Gould-Hsu inversions, the Carlitz inversions and the Grüss inequality.
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page2of 14 Go Back Full Screen
Close
Contents
1 Introduction and Some Known Results 3
2 A New Inequality 6
3 Aq-Analogue of the Inequality 9
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page3of 14 Go Back Full Screen
Close
1. Introduction and Some Known Results
q-series, which are also called basic hypergeometric series, plays a very important role in many fields, such as affine root systems, Lie algebras and groups, number theory, orthogonal polynomials and physics, etc. In this paper, first we establish an inequality by means of the Gould-Hsu inversions, and then we obtain aq-analogue of the inequality.
We first state some notations and known results which will be used in the next sections. It is supposed in this paper that 0 < q < 1. The q-shifted factorial is defined by
(1.1) (a;q)0 = 1, (a;q)n=
n−1
Y
k=0
(1−aqk), (a;q)∞ =
∞
Y
k=0
(1−aqk).
Theq-binomial coefficient is defined by
(1.2) hn
k i
= (q;q)n (q;q)k(q;q)n−k
.
The following inverse series relations are due to Gould-Hsu [4]:
Theorem 1.1. Let {ai} and {bj} be two real or complex sequences such that the polynomials defined by
ψ(x, n) =
n−1
Q
k=0
(ak+xbk), (n = 1,2, . . .), ψ(x,0) = 1,
differ from zero for any non-negative integerx. Then we have the following inverse
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page4of 14 Go Back Full Screen
Close
series relations
(1.3)
f(n) =
n
P
k=0
(−1)k nk
ψ(k, n)g(k), g(n) =
n
P
k=0
(−1)k nk ak+kbk
ψ(n, k+1)f(k), where nk
= k!(n−k)!n! .
Carlitz [2] gave the followingq-analogue of the Gould-Hsu inverse series rela- tions:
Theorem 1.2. Let {ai} and {bj} be two real or complex sequences such that the polynomials defined by
φ(x, n) =
n−1
Q
k=0
(ak+qxbk), (n= 1,2, . . .), φ(x,0) = 1,
differ from zero forx = qn with n being non-negative integers. Then we have the following inverse series relations
(1.4)
f(n) =
n
P
k=0
(−1)kn
k
q(n−k2 )
φ(k, n)g(k);
g(n) =
n
P
k=0
(−1)kn
k
ak+qkbk
φ(n;k+1)f(k).
We also need the following inequality, which is well known in the literature as the Grüss inequality [5]:
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page5of 14 Go Back Full Screen
Close
Theorem 1.3. We have (1.5)
1 b−a
Z b a
f(x)g(x)dx− 1
b−a Z b
a
f(x)dx 1
b−a Z b
a
g(x)dx
≤ (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 allx∈[a, b], wherem, M, n, N are given constants.
The discrete version of the Grüss inequality can be stated as:
Theorem 1.4. Ifa ≤ai ≤Aandb≤bi ≤Bfori= 1,2, . . . , n, then we have (1.6)
1 n
n
X
i=1
aibi− 1 n
n
X
i=1
ai· 1 n
n
X
i=1
bi
≤ (A−a)(B−b)
4 ,
wherea,A,ai,b,B,bi are real numbers.
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page6of 14 Go Back Full Screen
Close
2. A New Inequality
In this section we obtain an inequality about series by using both the Gould-Hsu inversions and the Grüss inequality.
Theorem 2.1. Suppose0≤a≤ f(k)≤A,g(k) =Pk i=0
k i
f(i), k = 1,2, . . . , n, then the following inequality holds
(2.1)
(n+ 1)
n
X
k=0
(−1)n+k n
k 2
f(k)g(k)−f(n)g(n)
≤3(n+ 1)22n−3A n
k0 A n
k0
−a
, wherek0 = [n−12 ],[x]denotes the greatest integer less than or equalx. Proof. Lettingai =−1, bi = 0in (1.3), we have
(2.2)
f(n) =
n
P
k=0
(−1)n+k nk g(k), g(n) =
n
P
k=0 n k
f(k).
Since0≤a≤f(k)≤A, we obtain a·
k
X
i=0
k i
≤g(k) =
k
X
i=0
k i
f(i)≤A·
k
X
i=0
k i
. SubstitutingPk
i=0 k
i
= 2kinto the above inequality we get (2.3) a·2k ≤g(k)≤A·2k, k= 0,1, . . . , n.
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page7of 14 Go Back Full Screen
Close
On the other hand, we know that
n k+1
n k
= n!/(k+ 1)!(n−k−1)!
n!/(k)!(n−k)! = n−k k+ 1, consequently
(k+1n )
(nk) ≥1 whenk ≤k0, (k+1n )
(nk) ≤1, whenk ≥k0, wherek0 = [n−12 ]. So, we get
(2.4) 1≤
n k
≤ n
k0
, k = 0,1, . . . , n.
LetAk= nk
f(k)andBk = (−1)n+k nk
g(k), then
(2.5) a≤Ak ≤A
n k0
. From (2.3) and (2.4), we know that
0≤Bk ≤2nA kn
0
ifn−kis even,
−2n−1A kn
0
≤Bk≤0, ifn−kis odd.
So, fork = 1,2, . . . , n, we have
(2.6) −2n−1A
n k0
≤Bk≤2nA n
k0
.
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page8of 14 Go Back Full Screen
Close
Combining (1.6), (2.5) and (2.6) we obtain
1 n+ 1
n
X
i=0
AiBi− 1 n+ 1
n
X
i=0
Ai
!
· 1 n+ 1
n
X
i=0
Bi
!
≤
A kn
0
−a 2nA kn
0
+ 2n−1A kn
0
4 ,
which can be written as
1 n+ 1
n
X
k=0
(−1)n+k n
k 2
f(k)g(k)
− 1 n+ 1
n
X
k=0
n k
f(k)
!
· 1 n+ 1
n
X
k=0
(−1)n+k n
k
g(k)
!
≤
A kn
0
−a 2nA kn
0
+ 2n−1A kn
0
4 .
Substituting (2.2) into the above inequality, we get (2.1).
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page9of 14 Go Back Full Screen
Close
3. A q-Analogue of the Inequality
In this section we give aq-analogue of the inequality (2.1) by means of the Carlitz inversions. First, we have the following lemma.
Lemma 3.1. Suppose 0 ≤ f(k) ≤ A and g(k) = Pk i=0
k
i
f(i), then for any k = 1,2, . . . , n, we have
(3.1) 0≤g(k)≤A
n
X
i=0
hn i i
.
Proof. It is obvious thatg(k)≥0. Ifk ≤n1 ≤n2, then we have hn2
k i
= 1−qn1+1
1−qn1+1−k · 1−qn1+2
1−qn1+2−k· · · 1−qn2 1−qn2−k
hn1 k
i . Since
1−qn1+1
1−qn1+1−k · 1−qn1+2
1−qn1+2−k · · · 1−qn2 1−qn2−k ≥1, we get
hn2 k
i≥hn1 k
i . Consequently,
g(k) =
k
X
i=0
k i
f(i)≤
k
X
i=0
hn i i
f(i)≤
n
X
i=0
hn i i
f(i).
The main result of this section is the following theorem.
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page10of 14 Go Back Full Screen
Close
Theorem 3.2. Suppose0≤a≤f(k)≤A,g(k) =Pk i=0
k
i
f(i),k = 1,2, . . . , n, then the following inequality holds
(3.2)
(n+ 1)
n
X
k=0
(−1)n+khn i
i2
q(n−k2 )f(k)g(k)−f(n)g(n)
≤ A(n+ 1)2 4
n k0
A n
k0
−a n
X
i=0
hn i i
+
n−1
X
i=0
n−1 i
! , wherek0 = [n−12 ],[x]denotes the greatest integer less than or equalx.
Proof. Lettingai =−1, bi = 0in (1.4) we get
(3.3)
f(n) =
n
P
k=0
(−1)n+kn
k
q(n−k2 ) g(k), g(n) =
n
P
k=0
n
k
f(k).
Using the lemma, we have
(3.4) a·
k
X
i=0
k i
≤g(k) =
k
X
i=0
k i
f(i)≤A·
n
X
i=0
hn i i
. On the other hand, we notice that
n
k+1
n
k
= (q;q)n/(q;q)k+1(q;q)n−k−1
(q;q)n/(q;q)k(q;q)n−k
= 1−qn−k 1−qk+1,
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page11of 14 Go Back Full Screen
Close
consequently
[k+1n ]
[nk] ≥1, whenk ≤k0, [k+1n ]
[nk] ≤1, whenk ≥k0, wherek0 = [n−12 ]. So, we have
(3.5) 1≤hn
k i ≤
n k0
, k = 0,1, . . . , n.
LetAk=n
k
f(k)andBk = (−1)n+kn
k
q(n−k2 )
g(k), then
(3.6) a ≤Ak≤A
n k0
. From (3.4) and (3.5), we know that
0≤Bk ≤Ah
n k0
i n P
i=0
n
i
, ifn−kis even,
−Ah
n k0
in−1 P
i=0
n−1 i
≤Bk ≤0, ifn−kis odd.
So, fork = 1,2, . . . , n, we get
(3.7) −A
n k0
n−1 X
i=0
n−1 i
≤Bk ≤A n
k0 n
X
i=0
hn i i
.
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page12of 14 Go Back Full Screen
Close
Combining (1.6), (3.6) and (3.7) we obtain
1 n+ 1
n
X
i=0
AiBi− 1 n+ 1
n
X
i=0
Ai
!
· 1 n+ 1
n
X
i=0
Bi
!
≤ 1 4
A
n k0
−a
A n
k0 n
X
i=0
hn i i
+A n
k0 n−1
X
i=0
n−1 i
! , which can be written as
1 n+ 1
n
X
k=0
(−1)n+khn k
i2
q(n−k2 )f(k)g(k)
− 1 n+ 1
n
X
k=0
hn k i
f(k)
! 1 n+ 1
n
X
k=0
(−1)n+khn k i
q(n−k2 ) g(k)
!
≤ A 4
n k0 A
n k0
−a n
X
i=0
hn i i
+
n−1
X
i=0
n−1 i
! . Substituting (3.3) into the above inequality, we get (3.2).
From [3], we know
limq→1
hn i i
= n
i
. Letq →1in both sides of the inequality (3.2) to get
(n+ 1)
n
X
k=0
(−1)n+k n
k 2
f(k)g(k)−f(n)g(n)
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page13of 14 Go Back Full Screen
Close
≤ A(n+ 1)2 4
n k0 A
n k0
−a " n
X
i=0
n 2
+
n−1
X
i=0
n−1 2
#
= A(n+ 1)2 4
n k0
A n
k0
−a
[2n+ 2n−1] = 3(n+ 1)22n−3A n
k0
A n
k0
−a
, which is the inequality (2.1). So the inequality (3.2) is theq-analogue of the inequal- ity (2.1).
An Inequality and its q-Analogue Mingjin Wang vol. 8, iss. 2, art. 50, 2007
Title Page Contents
JJ II
J I
Page14of 14 Go Back Full Screen
Close
References
[1] G.E. ANDREWS, R. ASKEYANDR. ROY, Special Functions, Cambridge Uni- versity Press, 2000.
[2] L. CARLITZ, Some inverse relations, Duke Math. J., 40 (1973), 893–901 [3] G. GASPERANDM. RAHMAN, Basic Hypergeometric Series, Encyclopedia of
Mathematics and Its Applications, 35, Cambridge University Press, Cambridge and New York, 1990.
[4] H.W. GOULDAND L.C. HSU, Some new inverse series relations, Duke Math.
J., 40 (1973), 885–891
[5] G. GRÜSS, Über das maximum des absoluten Betrages von
1 b−a
Rb
a f(x)g(x)dx − (b−a1 Rb
af(x)dx)(b−a1 Rb
a g(x)dx), Math.Z., 39 (1935), 215–226.
[6] 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].