Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 705168, 9 pages
doi:10.1155/2010/705168
Research Article
A Generalization of the Cauchy-Schwarz Inequality
with Eight Free Parameters
Mohammad Masjed-Jamei
Department of Mathematics, K.N.Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran
Correspondence should be addressed to Mohammad Masjed-Jamei, mmjamei@kntu.ac.ir
Received 6 September 2009; Accepted 10 February 2010
Academic Editor: Sever Silvestru Dragomir
Copyright q 2010 Mohammad Masjed-Jamei. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The results of the recent published paper by Masjed-Jamei et al. 2009 are extended to a larger
class and some of subclasses are studied in the sequel. In other words, we generalize the well
known Cauchy-Schwarz and Cauchy-Bunyakovsky inequalities having eight free parameters and
then introduce some of their interesting subclasses.
1. Introduction
Let {ak }nk1 and {bk }nk1 be two sequences of real, numbers. It is well known that the discrete
version of the Cauchy-Schwarz inequality 1, 2 can be stated as
n
2
≤
ak bk
k1
n
k1
a2k
n
bk2 ,
1.1
k1
while its continuous version in the space of real-valued functions Ca, b, R, that is, the
Cauchy-Bunyakovsky inequality 1, 2, can be represented as
2
b
fxgxdx
a
≤
b
a
f 2 xdx
b
g 2 xdx.
1.2
a
To date, a large number of generalizations and refinements of inequalities 1.1 and 1.2 have
been investigated in the literature e.g., 3–6. In this sense, recently in 7 a generalization of
the Cauchy-Bunyakovsky-Schwarz inequality with four free parameters has been presented
as follows.
2
Journal of Inequalities and Applications
Theorem 1.1. If {ak }nk1 and {bk }nk1 are two sequences of real numbers and p, q, r, s belong to a
subset of R, then we have
⎛
⎝
n
ak bk A1
k1
n
ak
k1
n
bk B1
n
k1
2
C1
ak
n
k1
2 ⎞2
bk ⎠
k1
⎛
2
2 ⎞
n
n
n
n
n
≤ ⎝ a2k A2
ak
bk B2
ak
C2
bk ⎠
k1
k1
k1
k1
1.3
k1
⎛
2
2 ⎞
n
n
n
n
n
× ⎝ bk2 A3
ak
bk B3
ak
C3
bk ⎠ ,
k1
k1
k1
k1
k1
in which
⎛
⎞
⎞
⎛
A1 B1 C1
p s ps qr r 1 p q 1 s
⎜
⎟ 1⎜
⎟
⎟
⎜
M0 ⎜
p p2
q2 ⎟
⎝A2 B2 C2 ⎠ n ⎝ 2q 1 p
⎠.
A3 B3 C3
2r 1 s
r2
s s 2
1.4
The inequality 1.3 generalizes the Cauchy-Schwarz inequality for Ai Bi Ci 0 i 1, 2, 3 and
the equality holds if ak bk and Ai Bi Ci for each i 1, 2, 3.
Moreover, if f, g : a, b → R are two integrable functions on a, b and p, q, r, s belong to a
subset of R, then the continuous version of inequality 1.3 reads as
⎛
⎝
b
fxgxdx a
⎛
≤⎝
b
a
⎛
×⎝
A∗1
b
b
fxdx
a
f 2 xdx A∗2
b
a
gxdx a
b
b
fxdx
g 2 xdx A∗3
b
gxdx B2∗
a
a
b
b
fxdx
a
B1∗
a
2
fxdx
a
b
gxdx B3∗
C1∗
b
a
2
fxdx
a
b
C2∗
b
2
fxdx
a
2 ⎞2
gxdx ⎠
C3∗
2 ⎞
gxdx ⎠
a
b
2 ⎞
gxdx ⎠,
a
1.5
in which
⎛
⎞
p s ps qr r 1 p q 1 s
⎟
⎟
⎜
1
⎜
p p2
q2 ⎟
C2∗ ⎟
⎠ b − a ⎝ 2q 1 p
⎠.
∗
2
C3
2r1 s
r
ss 2
A∗1 B1∗ C1∗
⎜ ∗ ∗
M0∗ ⎜
⎝ A2 B 2
A∗3 B3∗
⎞
⎛
1.6
The inequality 1.5 generalizes the Cauchy-Bunyakovsky inequality for A∗i Bi∗ Ci∗ 0 i 1, 2, 3 and the equality holds if fx gx and A∗i Bi∗ Ci∗ for each i 1, 2, 3.
Journal of Inequalities and Applications
3
The main aim of this paper is to generalize the abovementioned theorem by means of
eight free parameters and then study some of its particular cases of interest.
Theorem 1.2. Let {ak }nk1 and {bk }nk1 be two sequences of real numbers and {pi , qi }4i1 belong to a
subset of R. Then the following inequality holds
⎛
2
2 ⎞2
n
n
n
n
n
n
n
⎝A1 ak bk A2 ak bk A3 a2 A4
ak
A5 bk2 A6
bk ⎠
k
k1
k1
k1
k1
k1
k1
k1
2
2 ⎞
n
n
n
n
n
n
n
≤ ⎝B1 ak bk B2 ak bk B3
a2k B4
ak
B5 bk2 B6
bk ⎠
⎛
k1
⎛
× ⎝C1
n
k1
k1
ak bk C2
n
k1
ak
k1
k1
n
k1
bk C3
n
k1
a2k C4
k1
n
k1
2
ak
k1
C5
k1
n
2 ⎞
n
bk2 C6
bk ⎠ ,
k1
k1
1.7
in which
A1 p1 q2 p2 q1 ,
A2 p4 q1 nq3 q4 p1 np3 p2 q3 p3 q2 ,
A3 p1 q1 ,
A4 p1 q3 p3 q1 np3 q3 ,
A5 p2 q2 ,
A6 p2 q4 p4 q2 np4 q4 ,
B1 2p1 p2 ,
B2 2p4 p1 np3 2p2 p3 ,
B3 p12 ,
B4 p3 2p1 np3 ,
B5 p22 ,
B6 p4 2p2 np4 ,
C1 2q1 q2 ,
C2 2q4 q1 nq3 2q2 q3 ,
C3 q12 ,
C4 q3 2q1 nq3 ,
C5 q22 ,
C6 q4 2q2 nq4 .
1.8
4
Journal of Inequalities and Applications
Moreover, the equality in 1.7 holds if Ai Bi Ci for i 1, 2, . . . , 6, which is equivalent to
{pi qi }4i1 .
Proof. The proof of this theorem is straightforward if one defines a positive quadratic
polynomial P2 : R → R as
P2 x ; {pi , qi }4i1 n
⎛⎛
⎝⎝p1 ak p2 bk p3
n
aj p4
j1
k1
⎛
⎝q1 ak q2 bk q3
n
j1
aj q4
n
⎞
bj ⎠ x
j1
n
1.9
⎞⎞2
bj ⎠⎠ ,
j1
in which {ak }nk1 and {bk }nk1 are real numbers and {pi , qi }4i1 belong to a subset of R. Because
a simple calculation first reveals that
P2 x ; {pi , qi }4i1
⎛
⎞2
n
n
n
⎝p1 ak p2 bk p3
aj p4
bj ⎠ x2
j1
k1
2x
n
⎛
⎝p1 ak p2 bk p3
j1
n
j1
k1
n
⎛
⎝q1 ak q2 bk q3
k1
aj p4
n
n
j1
⎞⎛
bj ⎠⎝q1 ak q2 bk q3
j1
aj q4
n
n
j1
aj q4
n
⎞
bj ⎠
j1
⎞2
bj ⎠
j1
≥ 0,
1.10
therefore, for any x ∈ R, the discriminant Δ of P2 must be negative, that is,
⎛ ⎛
⎞⎛
⎞⎞2
n
n
n
n
n
1
Δ ⎝ ⎝p1 ak p2 bk p3
aj p4
bj ⎠⎝q1 ak q2 bk q3
aj q4
bj ⎠⎠
4
j1
j1
j1
j1
k1
⎛ ⎛
⎞2 ⎞
n
n
n
⎟
⎜
− ⎝ ⎝p1 ak p2 bk p3
aj p4
bj⎠ ⎠
k1
j1
j1
⎛ ⎛
⎞2 ⎞
n
n
n
⎟
⎜
aj q4
bj⎠ ⎠
× ⎝ ⎝q1 ak q2 bk q3
k1
j1
j1
≤ 0.
1.11
Journal of Inequalities and Applications
5
On the other hand, the elements of Δ/4 can eventually be simplified as follows:
⎛
⎞⎛
⎞
n
n
n
n
n
⎝p1 ak p2 bk p3
aj p4
bj ⎠⎝q1 ak q2 bk q3
aj q4
bj ⎠
j1
k1
A1
n
n
ak bk A2
k1
n
ak
k1
⎛
⎝p1 ak p2 bk p3
n
n
A3
n
ak bk A4
k1
bk B1
⎝q1 ak q2 bk q3
n
A5
n
ak bk A6
k1
a2k B2
n
j1
2
C1
ak
k1
n
bk2 C2
n
k1
2
bk
,
k1
⎞2
bj ⎠
j1
n
ak
n
n
bk B3
k1
n
n
a2k B4
n
k1
aj q4
j1
k1
n
k1
aj p4
k1
⎛
j1
k1
j1
k1
n
j1
n
2
C3
ak
k1
n
bk2 C4
n
k1
2
bk
,
k1
⎞2
bj ⎠
j1
ak
k1
n
bk B5
k1
n
a2k B6
n
k1
2
C5
ak
k1
n
bk2 C6
n
k1
2
bk
,
k1
1.12
where {Ai , Bi , Ci }6i1 are the same values as given in 1.8.
Hence, replacing the results 1.12 into 1.11 proves the first part of Theorem 1.2. The
proof of second part is also clear.
Similarly, one can consider the continuous analogue of Theorem 1.2 as follows.
Theorem 1.3. Let f, g : a, b → R be two integrable functions on a, b and {pi , qi }4i1 belong to a
subset of R. Then, the following inequality holds
⎛
⎝A∗
1
b
fxgxdx a
A∗5
b
a
⎛
≤
⎝B ∗
1
A∗2
b
g 2 xdx A∗6
b
b
fxdx
a
b
fxgxdx a
B5∗
b
a
gxdx a
A∗3
b
f xdx 2
a
A∗4
b
2
fxdx
a
2 ⎞2
gxdx ⎠
a
B2∗
g 2 xdx B6∗
b
b
fxdx
a
gxdx a
b
a
2 ⎞
gxdx ⎠
B3∗
b
a
f xdx 2
B4∗
b
2
fxdx
a
6
Journal of Inequalities and Applications
⎛
×
⎝C ∗
1
b
fxgxdx a
C5∗
b
a
C2∗
g xdx 2
b
b
fxdx
C6∗
gxdx a
a
b
2 ⎞
gxdx ⎠,
C3∗
b
f xdx 2
a
C4∗
b
2
fxdx
a
a
1.13
in which
A∗1 p1 q2 p2 q1 ,
A∗2 p4 q1 h∗ q3 q4 p1 h∗ p3 p2 q3 p3 q2 ,
A∗3 p1 q1 ,
A∗4 p1 q3 p3 q1 h∗ p3 q3 ,
A∗5 p2 q2 ,
A∗6 p2 q4 p4 q2 h∗ p4 q4 ,
B1∗ 2p1 p2 ,
B2∗ 2p4 p1 h∗ p3 2p2 p3 ,
B3∗ p12 ,
B4∗ p3 2p1 h∗ p3 ,
1.14
B5∗ p22 ,
B6∗ p4 2p2 h∗ p4 ,
C1∗ 2q1 q2 ,
C2∗ 2q4 q1 h∗ q3 2q2 q3 ,
C3∗ q12 ,
C4∗ q3 2q1 h∗ q3 ,
C5∗ q22 ,
C6∗ q4 2q2 h∗ q4 ,
so that h∗ b − a. Moreover, the equality in 1.13 holds if A∗i Bi∗ Ci∗ for i 1, 2, . . . , 6, which is
equivalent to {pi qi }4i1 .
Journal of Inequalities and Applications
7
Proof. The proof of this theorem is clearly similar to Theorem 1.2 if a positive quadratic
polynomial Q2 : R → R is defined as
Q2 x
; {pi , qi }4i1
b p1 ft p2 gt p3
b
a
fudu p4
a
q1 ft q2 gt q3
b
fudu q4
a
b
gudu
x
a
1.15
2
b
gudu
dt ≥ 0,
a
and then other items are followed accordingly.
2. Some Special Cases and a Unified Approach
As we observe in Theorems 1.2 and 1.3, there exist various subclasses of inequalities 1.7
and 1.13. In this section, we consider two of them. Naturally other cases can be studied
separately.
2.1. Special Case 1: Theorem 1.1, Inequalities 1.3 and 1.5
For example, one can conclude that inequality 1.3 is a special case of inequality 1.7 for
A1 1,
A2 1
p s ps qr ,
n
A3 0,
A4 1 r 1p ,
n
A5 0,
A6 1
q1 s,
n
2.1
where {Ai }6i1 are the same values as given in 1.8. Hence, after solving the above system,
one finally gets
p1 , p2 , p3 , p4 , q1 , q2 , q3 , q4 p q
r s
.
1, 0, , , 0, 1, ,
n n
n n
2.2
This means that replacing 2.2 in 1.7 gives the same as inequality 1.3.
2.2. Special Case 2: A Generalization of Pre-Grüss and Wagner Inequalities
It is clear that for
p1 , p2 , p3 , p4 , q1 , q2 , q3 , q4
p
s
,
1, 0, , 0, 0, 1, 0,
n
n
2.3
8
Journal of Inequalities and Applications
we obtain the following inequality:
2
n
n
p s ps ak bk ak
bk
n
k1
k1
k1
⎛
2 ⎞⎛
2 ⎞
n
n
n
n
p
p
2
ss
2
⎠.
≤ ⎝ a2k ak ⎠⎝ bk2 bk
n
n
k1
k1
k1
k1
n
2.4
Now, this inequality is a generalization of Pre-Grüss inequality 2 for p s −1 and is a
generalization of Wagner inequality 1, 8 for p s and pp 2/n w.
As we said, many special cases of inequalities 1.7 and 1.13 could be considered
independently. Hence, it is necessary that one classifies them by means of a unified approach.
For this purpose, let us reconsider two specific matrices M and M∗ defined by 1.8 and
1.14. It is clear that both matrices can directly determine all elements of inequalities 1.7
and 1.13, respectively. As a sample, substituting the arbitrary and random vector
p1 , p2 , p3 , p4 , q1 , q2 , q3 , q4 1, 2, 3, 4, 5, 6, 7, 8,
2.5
in the general matrix M gives
⎤
⎡
16 52n 60 5 21n 22 12 32n 40
⎥
⎢
⎥
M⎢
⎣ 4 24n 20 1 9n 6 4 16n 16⎦.
60 112n 164 25 49n 70 36 64n 96
2.6
Therefore, the inequality corresponding to 1.7 takes the form
⎛
⎝16
n
ak bk 52n 60
k1
n
ak
k1
12
⎛
≤ ⎝4
n
bk2 32n 40
k1
n
⎛
n
n
a2k
21n 22
k1
n
2
ak
k1
2 ⎞2
bk ⎠
k1
ak bk 24n 20
n
ak
k1
n
bk2 16n 16
k1
× ⎝60
bk 5
k1
k1
4
n
bk k1
n
2 ⎞
bk ⎠
n
a2k
9n 6
k1
n
2
ak
k1
2.7
k1
n
ak bk 112n 164
k1
36
n
n
k1
n
k1
bk2 64n 96
n
k1
ak
n
k1
bk 25
2 ⎞
bk ⎠.
n
k1
a2k
49n 70
n
k1
2
ak
Journal of Inequalities and Applications
9
Conversely, for instance the given inequality 1.5 has a characteristic matrix as
⎤
⎡
1 h∗ p s ps qr 0 h∗ r 1 p 0 h∗ q1 s
⎥
⎢
2h∗ q 1 p
M∗ Ineq.1.5 ⎢
1 h∗ p p 2 0
h∗ q2 ⎥
⎦.
⎣0
0
2h∗ r1 s
0
h∗ r 2
1 h∗ ss 2
2.8
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