Some generalizations of the Cauchy–Schwarz and the Cauchy–Bunyakovsky inequalities involving four free parameters and their applications

Mathematical and Computer Modelling 49 (2009) 1960–1968

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Some generalizations of the Cauchy–Schwarz and the Cauchy–Bunyakovsky inequalities involving four free parameters and their applications Mohammad Masjed-Jamei a , Sever S. Dragomir b , H.M. Srivastava c,∗ a

Department of Applied Mathematics, K. N. Toosi University of Technology, P. O. Box 1618, Tehran 16315–1618, Iran

b

School of Engineering and Science, Victoria University, P. O. Box 14428, Melbourne City MC, Victoria 8001, Australia

c

Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada

article

a b s t r a c t

info

Article history: Received 5 July 2008 Accepted 5 September 2008

Some generalizations of the well-known Cauchy–Schwarz inequality and the analogous Cauchy–Bunyakovsky inequality involving four free parameters are given for both discrete and continuous cases. Several particular cases of interest are also analyzed. Some of the applications of our main results include (for example) the Wagner inequality. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Generalization and refinement of the Cauchy–Schwarz inequality Wagner inequality Cauchy–Bunyakovsky inequality

1. Introduction Let {ak }nk=1 and {bk }nk=1 be two sequences of real numbers. It is well known that the discrete version of the Cauchy–Schwarz inequality [1] can be stated as follows: n X

!2 ak b k

k=1

5

n X

! a2k

k=1

n X

! b2k

.

(1.1)

k=1

The equality holds true in (1.1) if and only if the sequences are proportional, that is, if and only if ak = rbk

(k ∈ {1, . . . , n}),

(1.2)

where r ∈ R is a constant of proportionality. To date, a large number of generalizations and refinements of the Cauchy–Schwarz inequality (1.1) have been investigated in the literature (see, for example, the survey paper [2], the book [3] and the numerous references cited therein; see also several closely-related works such as [4–6]). In this paper, we present further generalizations of the Cauchy–Schwarz inequality (1.1) and the analogous Cauchy–Bunyakovsky inequality (4.1) below in terms of four free parameters and study its several particular cases of interest. One of the applications of our main results includes (for example) the Wagner inequality.

∗

Corresponding author. E-mail addresses: [email protected] (M. Masjed-Jamei), [email protected] (S.S. Dragomir), [email protected] (H.M. Srivastava). URLs: http://www.staff.vu.edu.au/rgmia/dragomir/ (S.S. Dragomir), http://www.math.uvic.ca/faculty/harimsri/ (H.M. Srivastava).

0895-7177/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2008.09.014

M. Masjed-Jamei et al. / Mathematical and Computer Modelling 49 (2009) 1960–1968

1961

2. A generalization of the Cauchy–Schwarz inequality Our first result is contained in Theorem 1. Theorem 1. If {ak }nk=1 and {bk }nk=1 are two sequences of real numbers and p, q, r , s ∈ R, then

 ! ! !2 !2 2 n n n n n X X X X X  ak bk + A1 ak bk + B1 ak + C1 bk  k=1

k=1

k=1

k=1

k=1

 ! ! !2 !2  n n n n n X X X X X 5 a2k + A2 ak bk + B2 ak + C2 bk  k=1

k=1

k=1

k=1

k=1

 ! ! !2 !2  n n n n n X X X X X b2k + A3 · ak bk + B3 ak + C3 bk  , k=1

k =1

k=1

k=1

(2.1)

k=1

in which the coefficients involved are defined by the following matrix equation: M =

A1 A2 A3

B1 B2 B3

C1 C2 C3

r (1 + p) p(p + 2) r2



p + s + ps + qr =  2q(1 + p) n 2r (1 + s)

!

1

q(1 + s) q2  . s(s + 2)



Moreover, the inequality (2.1) is equivalent to

 n X p + s + ps + qr  ak b k + n

k=1

5

n X

n X

a2k +

k=1

!

n X

ak

k=1

! +

bk

r (1 + p)

n X

n

k=1

!2 ak

+

q(1 + s)

k=1

n X

n

!2  2 bk 

k=1

!

! n n n n X X X X 1 [(p + 2)ak + qbk ] · b2k + [rak + (s + 2)bk ] , (2.2) (pak + qbk ) (rak + sbk )

n 1X

n k=1

k=1

n k =1

k=1

k=1

and is a generalization of the Cauchy–Schwarz inequality (1.1), which corresponds to the special case of (2.1) when Aj = Bj = Cj = 0 (j = 1, 2, 3). The equality holds true in (2.1) if ak = b k

(k ∈ {1, . . . , n}) and Aj = Bj = Cj (j = 1, 2, 3).

Proof. Let us define the positive quadratic polynomial Q :R→R as follows: Q (x; p, q, r , s) =

n X

" ak +

k=1

n pX

n k=1

ak +

n qX

n k=1

! x+

bk

bk +

n r X

n k=1

ak +

n s X

n k=1

!#2 ,

bk

(2.3)

in which p, q, r , s ∈ R, and {ak }nk=1 and {bk }nk=1 are real numbers. Since a simple calculation reveals that Q (x; p, q, r , s) =

n X

ak +

k =1

+2

n X

n pX

n k =1

ak +

k=1

+

n X k =1

bk +

ak +

n pX

n k=1

n r X

n k=1

n qX

n k=1

ak +

ak +

!2 x2

bk

n qX

n k=1

n s X

n k=1

! bk

bk +

n r X

n k=1

ak +

n s X

n k=1

! bk

x

!2 bk

=0

(x ∈ R),

(2.4)

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M. Masjed-Jamei et al. / Mathematical and Computer Modelling 49 (2009) 1960–1968

the discriminant ∆ of Q must be negative, that is, 1 4

" n X

∆=

ak +

k=1

−

n X

n pX

n k=1

ak +

k=1

n qX

ak +

n pX

n k=1

n k=1

!

ak +

bk +

bk

n qX

n k=1

n r X

n k=1

n s X

ak +

n k=1

!#2 bk

!2  !2  n n n X s X r X bk · bk + ak + bk  5 0 . n k=1

k=1

(2.5)

n k =1

On the other hand, the elements of ∆/4 can be simplified as follows: n X

ak +

k=1

=

n X

n pX

n k=1

ak b k +

ak +

n pX

ak +

n k=1

k=1

n k=1

!

n X

n

ak +

n r X

bk +

bk

p + s + ps + qr

k=1 n X

n qX

n k=1

!

n X

ak

k=1

n qX

n k=1

!2

n k=1

! bk

+

! bk

r (1 + p)

a2k

+

2q(1 + p)

+

p(p + 2) n

n X

k=1 2

!

+

ak

!2 +

ak

q(1 + s) n

k=1

n X

n

k=1

n X

n

k=1 n X

=

bk

ak +

n s X

k=1

!

n X

ak

n X

!2 bk

,

(2.6a)

k=1

! bk

k=1

q2

n X

n

!2 bk

(2.6b)

k=1

and n X

bk +

k=1

=

n X

b2k

n r X

n k=1

+

k=1

ak +

n s X

n k=1

2r (1 + s) n

!2 bk

n X

!

n X

ak

k=1

! bk

+

k=1

r2 n

n X

!2 ak

k=1

+

s(s + 2) n

n X

!2 bk

.

(2.6c)

k=1

So, upon substituting the results from (2.6) into the inequality (2.5), the first part of Theorem 1 is proved. To prove the second part (i.e., the equality condition), let us assume that b k = v ak

(k ∈ {1, . . . , n})

and substitute it into (2.1) to get

 v

n X

a2k

+ (C1 v + A1 v + B1 ) 2

=

n X

!2 2 ak 

k=1

k=1



n X

a2k + (C2 v 2 + A2 v + B2 )

k=1

n X

!2   !2  n n X X ak  · v 2 a2k + (C3 v 2 + A3 v + B3 ) ak  .

k=1

k=1

(2.7)

k=1

After some computations, the above equality leads us to the following nonlinear system:

(C1 v 2 + A1 v + B1 )2 = (C2 v 2 + A2 v + B2 )(C3 v 2 + A3 v + B3 ), 2v(C1 v 2 + A1 v + B1 ) = v 2 (C2 v 2 + A2 v + B2 ) + (C3 v 2 + A3 v + B3 ).



(2.8)

Obviously, one of the solutions of Eq. (2.8) is given by Aj = Bj = Cj

(j = 1, 2, 3) and v = 1.

Remark 1. There exist various sub-cases of the inequality (2.1). However, due to page limitations, here we only consider a particular case of (2.1) and investigate its sub-cases. Naturally, therefore, other special cases can be separately studied. The details involved are being left as an exercise for the interested reader. 3. The particular case when B1 = C1 = 0 A total of four cases can occur for the inequality (2.1) when B1 = C1 = 0.

M. Masjed-Jamei et al. / Mathematical and Computer Modelling 49 (2009) 1960–1968

1963

They are given, respectively, as follows: (i) (ii) (iii) (iv)

(r , q) = (0, 0); (r , s) = (0, −1); (p, q) = (−1, 0); (p, s) = (−1, −1).

3.1. The case when q = r = 0 and p, s ∈ R in (2.1) In this case, we have B1 = C1 = A2 = C2 = A3 = B3 = 0 and the inequality (2.1) is reduced to the following form:

"

n X

ak b k +

n X

p + s + ps n

k=1

! ak

k=1

n X

!#2 bk

k =1

 !2   !2  n n n n X X X X s ( s + 2 ) p ( p + 2 ) ak   b2k + bk  . 5 a2k + n

k=1

k=1

n

k=1

(3.1)

k=1

This inequality has some interesting sub-cases as detailed below. 3.1.1. Sub-Case 1. p = s ∈ R \ (−2, 0): (A Generalization of the Wagner Inequality) The following inequality for sequences of real numbers is known in the literature as the Wagner inequality [7] (see also [8]): Let {ak }nk=1 and {bk }nk=1 be two sequences of real numbers. If w = 0, then

"

n X

n X

ak b k + w

k=1

! ak

k=1

n X

!#2 bk

 !2   !2  n n n n X X X X 5 a2k + w ak   b2k + w bk  .

k=1

k =1

k=1

k=1

(3.2)

k=1

In order to deduce (3.2) from our results, it is sufficient to assume in (3.1) that p + s + ps n

=

p(p + 2) n

=

s(s + 2) n

= 0,

which obviously holds true for p = s ∈ R \ (−2, 0) and readily yields the Wagner inequality (3.2) for

w=

p(p + 2) n

= 0.

We note that, if in (3.1) we let p(p + 2) 5 0

and

s(s + 2) 5 0,

then

"

n X

ak b k +

p + s + ps

k=1

n

n X k=1

! ak

n X

!#2 bk

 !2   !2  n n n n X X X X p ( p + 2 ) s ( s + 2 ) 5  ak   b2k + bk  a2k +

k =1

n

k=1

5

n X k=1

! a2k

n X

k=1

k=1

n

k=1

! b2k

⇐⇒ p ∈ [−2, 0] and s ∈ [−2, 0].

(3.3)

k=1

3.1.2. Sub-Case 2. p = s ∈ [−2, 0] : (A Refinement for the Cauchy–Schwarz Inequality) Suppose in (3.1) that p = s ∈ [−2, 0]

and p(p + 2) = u.

Consequently, we have u ∈ [−1, 0]. By noting these assumptions, we can obtain a refinement for the Cauchy–Schwarz inequality (1.1). For this purpose, we first consider the following inequality, which is directly provable via some algebraic

1964

M. Masjed-Jamei et al. / Mathematical and Computer Modelling 49 (2009) 1960–1968

computations:

 !2   !2   ! 21 n n n n n X X X X X u u  a2k + ak   b2k + bk  5  a2k n

k =1

k=1

n

k=1

k=1

n X

k=1

! 12 b2k

+

k=1

n X

u n

! ak

k=1

n X

!2 bk  ,

(3.4)

k=1

which leads us eventually to



n X

u n



! 12 a2k

k=1

n X

n X

bk −

k=1

! 12 b2k

k=1

n X

2 ak  5 0 (u ∈ [−1, 0]).

(3.5)

k=1

Hence, by referring to the inequalities (3.1) and (3.4), we can at last arrive at the following corollary. Corollary 1. Let {ak }nk=1 and {bk }nk=1 be two positive sequences of real numbers and α ∈ [0, 1]. Then

" n X

ak bk −

k=1

α

n X

n

! ak

k=1

n X

!#2 bk

 n X α a2k − 5  n

k=1

k=1

 5 

n X

n X

!2   n X α b2k − ak  

! 12

n X

a2k

! 12 b2k

−

k=1

k=1

n

k=1

k=1

α n

n X

n X

!2  bk 

k=1

! ak

k=1

n X

!2 bk  ,

(3.6)

k=1

which is equivalent to the following inequality: n X

!2 ak b k

 α 5  n

k=1

5

n X

n X

!

n X

ak

k=1

k=1

!

!

a2k

k=1

n X

b2k

! bk

+

v u n uX t k =1

a2k −

α n

n X

v !2 u n uX α t ak b2k −

n X

k=1

k=1

k =1

n

!2 bk

2  

.

(3.7)

k =1

The equality holds true in (3.7) when b k = λ ak

(k ∈ {1, . . . , n}),

where λ is a constant. For other refinements of the Cauchy–Schwarz inequality (1.1), we refer the reader to the earlier works [9,10].

3.1.3. Sub-Case 3 It may be interesting to add that, if 1 p

+

1 s

= −1 (p, s ∈ R \ {0}),

then (3.1) is reduced to the following form: n X

!2 ak b k

 !2   !2  n n n n X X X X p ( p + 2 ) s ( s + 2 ) ak  ·  bk  , 5 a2k + b2k +

k=1

k=1

n

k=1

k=1

which, for p = s = −2, yields the Cauchy–Schwarz inequality (1.1). 3.2. The case when r = 0, s = −1 and p, q ∈ R in (2.1) In this case, we have B1 = C1 = A3 = B3 = 0

n

k=1

(3.8)

M. Masjed-Jamei et al. / Mathematical and Computer Modelling 49 (2009) 1960–1968

1965

and the inequality (2.1) is reduced to the following form:

"

n X

ak b k −

k=1

!

n X

1 n

n X

ak

k=1

 !2  n n X 1 X 2 5  bk − bk 

!#2 bk

k =1

n

k=1

k=1

 !2 !2  n n n X X X 1 1 · a2k − ak + [(p + 1)ak + qbk ]  . n

k=1

n

k=1

(3.9)

k=1

However, since n X

a2k

−

k=1

1 n

n X

!2 = 0,

ak

(3.10)

k=1

the best option for p and q in (3.9) is when p = −1 and q = 0. Furthermore, we note that the above-mentioned third case, that is, p = −1,

q=0

r , s ∈ R,

and

gives us the same result as in (3.9). 3.3. The case when p = s = −1 and q, r ∈ R in (2.1) In this case, we have B1 = C1 = A2 = A3 = 0 and the inequality (2.1) is reduced to the following form:

"

n X

ak b k +

n qr − 1 X

n

k=1

!

!#2

n X

ak

k=1

bk

 !2 n n X q2 1 X 2 5  ak − ak +

k=1

n

k=1

 ·

n X

b2k

−

k=1

n

k=1 n X

1 n

n X

!2 +

bk

k=1

r2 n

k=1

!2  bk 

n X

!2  ak  .

(3.11)

k=1

An interesting case in (3.11) occurs when q = r = 1. We are thus led to the following inequality: n X



!2 ak bk

5

k =1

n X

1

a2k +

 n  X

n

k=1

!2 bk

k=1

−

 !2 !2    !2   n n n  X   X X 1 · . ak ak − bk b2k +   n  k=1 k=1 k=1 k=1

n X

(3.12)

4. A generalization of the Cauchy–Bunyakovsky inequality In a similar manner, the integral version of the Cauchy–Schwarz inequality (1.1), which is known in the literature as the Cauchy–Bunyakovsky inequality [11] and has the following form: b

Z

f (x)g (x)dx

2

b

Z

[f (x)] dx

5

a

b

Z

2

a

[g (x)]2 dx,

(4.1)

a

can also be generalized as follows. Theorem 2. Let f , g : [a, b] → R be two integrable functions on [a, b] and p, q, r , s ∈ R. Then the following inequality holds true:

"Z

b

f (x)g (x)dx + A1

∗

b

Z

a

f (x)dx a

"Z

b

[f (x)] dx + A2 2

5

∗

"Z

[g (x)] dx + A3 2

a

b

f (x)dx

∗

f (x)dx a

f (x)dx

2

∗

b

g (x)dx + B2

∗

g (x)dx + B3

∗

a

g (x)dx

+ C1

b

Z

f (x)dx

2

∗

2 #2

Z a

b

Z

2 #

g (x)dx

+ C2

a

b

Z

b

Z a

a

b

Z

b

Z a

Z

a

b

·

g (x)dx + B1

∗

a

Z

a

b

Z

a

b

f (x)dx

2

∗

b

Z

g (x)dx

+ C3

a

2 #

,

(4.2)

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M. Masjed-Jamei et al. / Mathematical and Computer Modelling 49 (2009) 1960–1968

in which A∗1 A∗2 A∗3

M∗ =

B∗1 B∗2 B∗3

C1∗ C2∗ C3∗

r (1 + p) p(p + 2) r2



p + s + ps + qr  2q(1 + p) = b−a 2r (1 + s)

!

1

q(1 + s) q2  . s(s + 2)



Moreover, the inequality (4.2) is equivalent to the following integral inequality:

"Z

b

f (x)g (x)dx +

b−a

a b

Z

b−a

a b

Z

b−a

a

a

g (x)dx +

r (1 + p)

Z

a

b

2

b

Z

b−a

a

[pf (x) + qg (x)]dx

f (x)dx

+

q(1 + s) b−a

a

b

Z

g (x)dx

2 #2

a

 [(p + 2)f (x) + qg (x)]dx

a

b

Z

1

[g (x)]2 dx +

·

f (x)dx

b

Z

b

Z

1

[f (x)]2 dx +

5

b

Z

p + s + ps + qr

[rf (x) + sg (x)]dx

b

Z

a

 [rf (x) + (s + 2)g (x)]dx ,

(4.3)

a

and would reduce to the Cauchy–Bunyakovsky inequality (4.1) in its special case when A∗j = B∗j = Cj∗ = 0 (j = 1, 2, 3). The equality holds true in (4.2) if f (x) = g (x)

(j = 1, 2, 3).

A∗j = B∗j = Cj∗

and

Although the proof is similar to the proof of Theorem 1, by defining the positive quadratic polynomial R(x; p, q, r , s) by R(x; p, q, r , s) =

b

Z



f (t ) +

a

 +

g (t ) +

b

Z

p b−a r

b−a

b−a

a b

Z

b−a

a



g (x)dx x a

b

Z

s

f (x)dx +

b

Z

q

f (x)dx +

g (x)dx

2

dt = 0,

(4.4)

a

we note that the following relations are required to be applied in our proof of Theorem 2: b

Z



f (x) +

a

b

Z

p b−a

f (x)g (x)dx +

=

p + s + ps + qr b−a

a

+ b

Z



r (1 + p) b−a

f (x) +

a

b

Z

f (x)dx

b−a

+

f (x)dx +

[f (x)]2 dx +

2q(1 + p)

b

f (x)dx

Z

a

s

b

Z

b−a



g (x)dx dx a

g (x)dx

g (x)dx

2

,

(4.5)

a

g (x)dx

2 dx

a

b

Z

b−a

f (x)dx +

b

b

Z

b

Z

b

Z

r

a

b−a

b−a

b−a

a

  · g (x) +

a

Z

q(1 + s) q

a

b

Z =

2

b

Z

g (x)dx

a

a

p

b

Z

b−a

a

b

Z

q

f (x)dx +

f (x)dx

b

Z

a

g (x)dx + a

p(p + 2)

Z

b−a

b

f (x)dx

2

a

+

q2

b

Z

b−a

g (x)dx

2 (4.6)

a

and b

Z



g (x) +

a

r b−a

b

Z

[g (x)]2 dx +

= a

b

Z

f (x)dx + a

2r (1 + s) b−a

s b−a

b

Z

g (x)dx

f (x)dx a

dx

a

b

Z

2

b

Z

g (x)dx + a

r2

b

Z

b−a

f (x)dx a

2 +

s(s + 2) b−a

b

Z

2

g (x)dx

,

(4.7)

a

successively. Furthermore, we note that all of the above-mentioned sub-cases for the inequality (2.1) can similarly be considered for the continuous (integral) case given by (4.2). For the sake of completeness, we can state the following corollary as one of the results derived in this manner. Corollary 2 (A Refinement of the Cauchy–Bunyakovsky Inequality). Let f , g : [a, b] → R

M. Masjed-Jamei et al. / Mathematical and Computer Modelling 49 (2009) 1960–1968

1967

be two positive integrable functions on the interval [a, b] and α ∈ [0, 1]. Then b

Z

f (x)g (x)dx



2

α b−a

5 

a

s Z

b

Z

f (x)dx

b

Z

[f (x)]2 dx

b

[f (x)]2 dx − a

α b−a

[g (x)]2 dx − a

5

g (x)dx +

s Z

a

a

b

·

b

Z

b

Z

g (x)dx

2

α b−a

b

Z

f (x)dx

2

a

2 

a

b

Z

[g (x)]2 dx.

(4.8)

a

a

5. A unified approach for the classification of (2.1) and (4.2) As we observed in the preceding sections, there are, respectively, two special matrices M and M ∗ for the inequalities (2.1) and (4.2) having 9 elements. Consequently, each sub-case of these two inequalities can be characterized by means of the matrices M or M ∗ directly. For instance, the following discrete inequality:

 ! ! n n n X X s+2 X r  ak b k − ak bk − n

k=1

5

n X

a2k

k=1

n X

k=1

b2k

+

k=1

n

k=1

n 1X

n X

n k=1

k=1

(rak + sbk )

n X

!2 2 ak 

k=1

! [rak + (s + 2)bk ] ,

(5.1)

which yields the Cauchy–Schwarz inequality (1.1) in its special case when r =0

and

s = −2,

has the characteristic matrix given by

 M [Ineq. (5.1)] =

1

−s − 2

−r

0 2r (1 + s)

0 r2



n



0 0 s(s + 2)

,

(5.2)

while the following continuous inequality:

"Z

b

f (x)g (x)dx +

a

Z 5

b−a

b

b

Z

p

f (x)dx a

b−a

a

g (x)dx +

b−a

b

[pf (x) + qg (x)]dx · a

b

Z

q

a

Z

1

[f (x)]2 dx +

b

Z

g (x)dx

2 #2

a

b

Z

[(p + 2)f (x) + qg (x)]dx a

b

 Z

 [g (x)]2 dx ,

(5.3)

a

corresponds to the matrix given by



p 2q(1 + p) M ∗ [Ineq. (5.3)] = b−a 0



0

1

p(p + 2) 0

q q2  . 0

(5.4)

Similarly, for the inequalities (3.2), (3.9), (3.11) and (3.12), we have M [Ineq. (3.2)] =

p(p + 2) n

1 0 0

0 1 0

0 0 , 1

!

 M [Ineq. (3.9)] =

1 n

−1 2q(1 + p) 0



qr − 1 M [Ineq. (3.11)] =  0 n 0 1

(5.5)



0 p(p + 2) 0 0 −1 r2

0 q2  , −1

(5.6)



0 q2  −1

(5.7)

1968

M. Masjed-Jamei et al. / Mathematical and Computer Modelling 49 (2009) 1960–1968

and



0 1 0 M [Ineq. (3.12)] =  n 0

0

−1 1



0 1  , −1

(5.8)

respectively. Finally, we mention that the usual Cauchy–Schwarz and Cauchy–Bunyakovsky inequalities correspond, respectively, to M =0

and M ∗ = 0,

which can be obtained for p = q = r = s = 0 (see also [12,13]). Acknowledgements The present investigation is supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. References [1] A.L. Cauchy, Cours d’Analyse del École Royale Polytechnique. I: Analyse Algébrique, Gauthier-Villars, Paris, 1821. [2] S.S. Dragomir, A survey on Cauchy–Bunyakovsky–Schwarz type discrete inequalities, J. Inequal. Pure Appl. Math. 4 (3) (2003) Article 63. [3] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Classical and New Inequalities in Analysis, Mathematics and its Applications (East European Series), vol. 61, Kluwer Acad. Publ., Dordrecht, Boston and London, 1993. [4] G.V. Milovanović (Ed.), Recent Progress in Inequalities, Kluwer Series on Mathematics and its Applications, vol. 430, Kluwer Acad. Publ., Dordrecht, Boston and London, 1998. [5] D.S. Mitrinović, Analytic Inequalities, Die Grundlehren der Mathematischen Wissenschaften, Band 165, Springer-Verlag, New York and Berlin, 1970. [6] Th. M. Rassias, H.M. Srivastava (Eds.), Analytic and Geometric Inequalities and Applications, Mathematics and its Applications, vol. 478, Kluwer Acad. Publ., Dordrecht, Boston and London, 1999. [7] S.S. Wagner, Notices Amer. Math. Soc. 12 (1965) 220. [8] P. Flor, Über eine Unglichung von S.S. Wagner, Elem. Math. 20 (1965) 136. [9] H. Alzer, A refinement of the Cauchy–Schwartz inequality, J. Math. Anal. Appl. 168 (1992) 596–604. [10] L. Zheng, Remark on a refinement of the Cauchy–Schwartz inequality, J. Math. Anal. Appl. 218 (1998) 13–21. [11] V.Y. Buniakowski, Sur quelques inegalites concernant les integrales aux differences finies, Mem. Accad. St. Petersburg (7) 1 (9) (1859) 1–18. [12] S.S. Dragomir, On the Cauchy–Buniakowsky–Schwarz inequality for sequences in inner product spaces, Math. Inequal. Appl. 3 (2000) 385–398. [13] W.L. Steiger, On a generalization of the Cauchy–Schwarz inequality, Amer. Math. Monthly 76 (1969) 815–816.