SOOCHOW JOURNAL OF MATHEMATICS
Volume 21, No. 4, pp. 413-426, October 1995
SOME MAPPINGS ASSOCIATED WITH
CAUCHY-BUNIAKOWSKI-SCHWARZ'S INEQUALITY
IN INNER PRODUCT SPACES
BY
S. S. DRAGOMIR AND B. MOND
1. Introduction
Let ( ( )) be an inner product space over the real or complex number
eld K. The following inequality is well known in the literature as CauchyBuniakowski-Schwarz's inequality:
H
X
i i
2 X p jjx jj2
i i
i2I
p j j
i2I
jj
X
i2I
i i i
2
p x jj
(1 1)
:
where i 0, i K, i
for all
and is a nite set of indices.
The case of equality holds in (1.1) for i 0 (
) i there exists a
vector 0
so that i = i 0 for all
.
Indeed, a simple calculation shows that:
X
2
0 12
i j i j
j i
i j 2I
X
1
2
2 2Re(
2
2
=2
i j i
j
i j j i) + j
i ]
i j 2I
p
2
x
2 H
i 2 I
I
p
x
2 H
x
p p jj x
p p
=
=
x
X
i j 2I
X
i2I
2
; x jj
j
2;
p p j j jjx jj
i i
i 2 I
i 2 I
j j jjx jj
i j i
>
;
X
i j 2I
x
x
i j Re(j xj
p p
2 X p jjx jj2 ; jj X p x jj2
i i
i i i
i2I
i2I
p j j
Received December 14, 1994.
AMS Subject Classication. 26D20, 46C05.
413
j j jjx jj
i i)
x
414
S. S. DRAGOMIR AND B. MOND
and thus the equality holds in (1.1) i i j =
exists a vector 0
so that i = i 0 for all
We shall use the following notations:
x
x
2 H
x
P
f (N) :
x
=
N is nite
(R) : = = ( i )i2N i R
+ (R) : = = ( )
i i2N i 0
(K) : = = ( i )i2N i K
( ) : = = ( i )i2N i
fI jI
j i
x
i 2 I
for all
.
i j 2 I
, i.e., there
g
N
N
N
N
By the use of these notations we can dene the following mapping associated with the Cauchy-Buniakowski-Schwarz inequality (1.1):
S
S
S
S H
S
fp
p
jp
2
fp
p
jp
f
j
fx
x
jx
f (N) S
:
P
given by
(
) :=
S I p x
X
i2I
for all
for all
for all
for all
2
2 H
i 2
g
i 2
g
i 2
g
i 2
g:
+ (R) S (K) S (H ) ! R
i i
2 X p jjx jj2 ; jj X p x jj2 :
i i
i i i
i2I
i2I
p j j
The main aim of this paper is to point out the fundamental properties of
this mapping. Some natural applications are also given.
2. Some Properties of The Mapping (
)
S I p x
The following theorem holds:
Theorem 2.1. Let ( ( )) be an inner product space and (
H
)
S I p x
the mapping de ned above. Then:
(i) For all
0 one has the inequality:
p q (
S I p
+
)
(
q x
i.e., the mapping (
0 one has:
(ii) If
)+ (
S I p x
S I q x
) is superadditive on
S I
x
S
) 0
(2 1)
:
+ (R)
p q (
)
S I p x
(
S I q x
) 0
(2 2)
:
SOME MAPPINGS ASSOCIATED WITH CAUCHY-BUNIAKOWSKI-SCHWARZ'S INEQUALITY 415
i.e., the mapping (
) is monotonic nondecreasing on
S I
x
S
+ (R).
Proof. We will use the following identity proved above:
1X
2
(
)=2
S I s x
i j
i j ; j xi jj
s s jj x
i j 2I
:
(i) By the use of the previous identity one has:
X
( +
) = 21 ( i + i )( j + j ) i j j i 2
i j 2I
X
2 1 X
= 12
i j i j
i j
j i +2
i j 2I
i j 2I
X
2 1 X
+ 21
j i +2
i j i j
j
i j 2I
i j 2I
S I p
q x
p
q
p
p p jj x
q
; x jj
+
) (
q x
X
)+ (
)] =
; S I p x
S I q x
2
i j ; j xi jj
i
p q jj x
)+
S I q x
from where we get
(
2
i j ; j xi jj
q q jj x
)+ (
S I p x
S I p
; x jj
; x jj
p q jj x
= (
jj x
i j 2I
i j
2
i j ; j xi jj
p q jj x
X
i j 2I
2
i j ; j xi jj
i j
p q jj x
0
i.e., the inequality (2.1).
(ii) Suppose that
0. Hence:
p q (
)= ( (
S I p x
S I
p ;q
)+
)
S I p ;q x
)
S I p ;q x
q x
(
)+ (
(
) 0
S I q x
)
from where we get
(
S I p x
)
(
;S I q x
and the statement (2.2) is proved.
Corollary 2.1.1. Let
for all
(K),
f (N) we have the inequality:
2 S
( ) and = ( i )i2N
x 2 S H
R. Then
I 2 P
X
2 X jjx jj2 ; jj X x jj2
i
i i
i2I
i2I
i2I
X 2 2 X 2 2
X
ji j sin i
jjxi jj sin i ; jj
(sin i )2 i xi jj2
i2I
X 2 2 i2IX 2 2 i2I X
+ ji j cos i jjxi jj cos i ; jj (cos i )2 i xi jj2
i2I
i2I
i2I
i
j j
0
416
S. S. DRAGOMIR AND B. MOND
which improves Cauchy-Buniakowski-Schwarz's inequality.
Corollary 2.1.2. Consider the set of sequences ( ) :=
S I
with 0
i1
p
for all
i 2
fp
N . Then one has the bound:
= ( i )i2N
p
g
2 X p jjx jj2 ; jj X p x jj2
i i
i i i
p2S (I ) i2I
i2I
i2I
X X
X
= ji j2 jjxi jj2 ; jj i xi jj2 :
i2I
i2I
i2I
0
X
sup i i
p j j
Another result of this type is embodied in the following theorem.
Theorem 2.2. With the above assumption, we can state:
f (N)
(i) For all
with
I J 2 P
(
S I J p x
= , one has the inequality:
I \ J
)
(
)+ (
S I p x
) 0
S J p x
(2 3)
:
for
0, i.e., the mapping (
) is superadditive as an index set
function on f (N)
(ii) If
one has:
f (N) with
p
S
p x
P
I J 2 P
I J
(
)
S I p x
for all
0, i.e., the mapping (
index set function on f (N).
p S
(
) 0
S J p x
p x
(2 4)
:
) is monotonic nondecreasing as an
P
(
Proof. (i) Suppose that
X
1
S I J p x
) =2
f (N)
I J 2 P
i j
i j
2
i j ; j xi jj
j i
2+ 1
j i
X
S I p x
as
X
(i j )2I J
i j
)+ (
S J p x
2
i j ; j xi jj
p p jj x
=
)+
X
(i j )2J I
2
j i
p p jj x ; x jj
p p jj x ; x jj
= (
i j
2 (i j)2J J i j
2+ 1 X
2 (i j)2J I i j
p p jj x ; x jj
i j
= . Then
I \ J
p p jj x
(i j )2(I J )(I J )
X
= 21
i j
(i j )2I I
X
+12
i j
(i j )2I J
with
i j
j i
2
p p jj x ; x jj
X
(i j )2I J
i j
i j
2
i j ; j xi jj
p p jj x
2
i j ; j xi jj
p p jj x
0
SOME MAPPINGS ASSOCIATED WITH CAUCHY-BUNIAKOWSKI-SCHWARZ'S INEQUALITY 417
and the inequality (2.3) is proved.
(ii) Suppose that
f (N) with
I J 2 P
(
) = ((
S I p x
S
I nJ
)
J p x
. Then
I J
)
(
)+ (
(
) 0
S I nJ p x
)
S J p x
from which we get
(
)
S I p x
(
)
;S J p x
and the inequality (2.4) is proved.
Now, let us denote:
n(
S
where
):= (
p x
I
n := f1
n
S I
2
:::
n
X
f (N)
n
n
2 X p jjx jj2 ; jj X p x jj2
i i
i i
i i i
i=1
i=1
i=1
)=
p x
ng 2 P
S I nJ p x
(
n 2
p j j
N
1).
n The following corollaries hold:
Corollary 2.2.1. With the assumptions of Theorem 2.2 we have:
"X
#
X
2X
2
2
0 sup
pi ji j
pi jjxi jj ; jj
pi i xi jj
I In i2I
i2I
i2I
n
n
n
X
X
X
= pi ji j2 pi jjxi jj2 ; jj pi i xi jj2 :
i=1
i=1
i=1
Corollary 2.2.2. With the above assumptions, one has:
n
n
n
X
X
2X
2
2
pi i xi jj
i=1
2
max fpi pj jji xj ; j xi jj g 0
1i<j n
i=1
i i
p j j
i=1
i
i
p jjx jj
; jj
which gives another type of re nement for Cauchy-Buniakowski-Schwarz's
inequality.
Also we have:
Corollary 2.2.3. The sequence
nondecreasing.
n
S
:=
n (p
S
)(
x
n 2
N) is monotonic
418
S. S. DRAGOMIR AND B. MOND
Now, if
(R) and : R R, then by ( ) we will denote the sequence
of real numbers ( i ) i2N .
With these terms, we have the following theorem:
p 2 S
f
ff p
!
f p
g
Theorem 2.3. With the above assumptions, we have:
(i) If : 0 ) 0
have the inequality:
f
(
1
( + (1
S I f tp
(ii) If : 0
(
))
1
!
( + (1
S I f tp
1
2
)
;t q
)
f
) is concave, then for all
!
x
(0
(
()
t S I f p
x
p q )2 (
) + (1
;t
t 2
()
S I f q
0 1] we
x
) (2 5)
:
) is logarithmically convex, i.e., ln is convex, then:
1
))
;t q
f
) (
x
()
S I f p
)]t (
x
S I f
t 2
()
S I f q
for all
0 and
0 1], i.e., the mapping (
cally convex on + (R).
p q 0 and
()
)]1;t
x
(2 6)
:
) is logarithmi-
x
S
Proof. (i) We have successively:
(
( + (1
S I f tp
1X
=2
i j 2I
))
;t q
( i + (1
)
x
) i) (
f tp
;t q
f tp
2
j + (1 ; t)qj )jji xj ; j xi jj
1 X ( ) + (1 ) ( )] ( ) + (1 ) ( )]
i
i
j
j
i j
j
2 i j 2I
X
X
= 12 2
( i ) ( j ) i j j i 2 + 12 (1 )2
( i) ( j )
i j 2I
i j 2I
tf p
t
f p
+ (1
t
2
;t f q
(
)
;t
X
i j 2I
()
t S I f p
f p
jj x
f q
) + (1
x
; t
2
i j ; j xi jj
jj x
; t
i.e., the inequality (2.5).
;t f q
; x jj
( i) ( j )
f p
tf p
)2 (
()
S I f q
)
x
jj x
f q
i
2
; x jj
f q
2
i j ; j xi jj
jj x
SOME MAPPINGS ASSOCIATED WITH CAUCHY-BUNIAKOWSKI-SCHWARZ'S INEQUALITY 419
(ii) We have, by Holder's inequality, that
( ( + (1 ) )
)
X
= 21
( i + (1 ) i ) ( j + (1 ) j ) i j
i j 2I
1 X ( )]t ( )]1;t ( )]t ( )]1;t
i j
i
j
j
2 i j2I i
X
= 21 ( i ) ( j )]t ( i ) ( j )]1;t i j j i 2
i j 2I
X
1
2 ( ( i ) ( j )]t )1=t i j j i 2 ]t
i j 2I
X
1
2 ( ( i ) ( j )]1;t ) 1;1 t i j j i 2 ]1;t
i j 2I
S I f tp
; t q
f tp
;t q
f p
t 2
f p
f p
f q
f p
f p
f q
f q
; t q
jj x
f q
f q
jj x
jj x
jj x
2
j i
; x jj
j i
2
; x jj
; x jj
; x jj
jj x
; x jj
= ( ( )
)]t ( ( )
)]1;t
(0 1). If = 0 or = 1, the inequality (2.6) also holds.
S I f p
for all
f tp
f q
f p
x
x
S I f q
t
x
t
This completes the proof of the theorem.
0
Corollary 2.3.1. Let
1
and be as above. Then for all : 0
) a concave function, we have the inequality:
( ( +2 )
) 41 ( ( )
)+ ( ( )
)] 0
S I f
p
q
x p
x
q
S I f p
f
x
S I f q
x
)
1
!
:
Remark 2.1. If ( ) = , by the above inequality we recapture the
f x
x
inequality (2.1).
Corollary 2.3.2. With the above assumptions, for : 0 ) 0 ) a
f
1
logarithmically convex function, one has the inequality:
)]2 ( ( )
) ( ()
( ( +2 )
S I f
p
q
x
S I f p
x S I f q
!
1
)
x :
The following inequality of Holder type also holds.
Theorem 2.4. Let ( ( )) be an inner product space and
H
2 S
(K) and
( ). Then for all
x 2 S H
(
) (
S I pq x
S I p
s
p q 0 one has the inequality
)]1=s (
x
S I q
t
x
)]1=t
f (N),
I 2 P
(2 7)
:
420
S. S. DRAGOMIR AND B. MOND
where
s 1 and 1 + 1 = 1. If
=s
=t
s <
1, the inequality (2 7) is reversed.
:
Proof. We will use the following Holder inequality for double sums, i.e.,
0
11=s 0
11=t
X
X
X
sA @
tA
@
i j 2I
where ij
Since
p
ij bij
a
ij aij bij
p
0 for all
i j 2
(
S I pq x
i j 2I
ij
=
ij aij
N and
X
) = 21
i j 2I
hence, choosing
p
p
i j ; j xi jj
jj x
2
p q p q jj x
2
41 X
2 i j2I
s s
i j
a
The case
s <
i
ij
=
i j
ij
p p
b
=
i j
q q
2
; x jj
i j
31=s 2
25 4 1 X
j i
2 i j2I
p p jj x ; x jj
x
2
i j ; j xi jj
i i j j
p q p q jj x
and
= ( s
)]1=s (
and the inequality (2.7) is proved.
S I p
ij bij
p
are as above.
s t
we get, by the above Holder inequality
X
(
) = 12
i i j j i j
j
i j 2I
S I pq x
i j 2I
S I q
t
x
t t
i j
i j
j i
31=t
25
q q jj x ; x jj
)]1=t
1 goes likewise and we will omit the details.
Corollary 2.4.1. With the above assumptions, one has the following
inequality of Cauchy-Buniakowski-Schwarz type:
)]2
(
S I pq x
(
S I p
2
) (
x S I q
2
)
x :
Now, let us observe that the inequality (2.1) can be improved in the following way:
Theorem 2.5. With the above assumptions, we have for
(
2 + q2
S I p
2 (
) (
) 0
x
S I pq x
; S I p
:
2
)+ (
x
S I q
2
p q )]
x
0, that:
SOME MAPPINGS ASSOCIATED WITH CAUCHY-BUNIAKOWSKI-SCHWARZ'S INEQUALITY 421
Proof. We have:
(
S I p
2 + q2
X
) = 21 ( 2i + i2 )( 2j + j2 )
x
p
i j 2I
q
p
q
1 X ( + )2
2 i j2I i j i j
X 22
= 12
i j i j
i j 2I
X 22
+ 12
i j i j
i j 2I
p p
q q
p p jj x
S I p
2
2+
j i
; x jj
X
i i j j
2
i j ; j xi jj
p q p q jj x
i j 2I
2
j i
; x jj
)+ (
x
2
i j ; j xi jj
jj x
q q jj x
= (
2
i j ; j xi jj
jj x
S I q
2
x
)+2 (
S I pq x
)
which gives the desired inequality.
Another result of this type is also embodied in the next theorem:
Theorem 2.6. With the above assumption, we have:
2
(
)+ (
S I p
x
S I q
2
x
)
(
;S I p
Proof. We have successively:
1X 2
2
2
(
S I p
;q
) =2
x
(
i j 2I
1 X(
2 i j2I
2
= (
S I p
p
i
2 ; q2
) 2 (
x
) 0
S I pq x
:
2 2
2
2
i )(pj ; qj )jji xj ; j xi jj
; q
2 jj x ; x jj2
i j
j i
i j ; qi qj )
p p
x
)+ (
S I q
2
) 2 (
x
;
)
S I pq x
from where we obtain the desired result.
Corollary 2.6.1. With the above assumptions, we have:
(
2 + q2
S I p
)
x
(
2 ; q2
;S I p
) 4 (
x
) 0
S I pq x
:
Further on, we shall give other results which contain renements of the
Cauchy-Buniakowski-Schwarz inequality:
422
S. S. DRAGOMIR AND B. MOND
Theorem 2.7. With the above assumptions, one has the following bound:
(
S I p x
where ( ) :=
S p
fq 2 S
Proof. Let
(R)
) = sup
q2S (p)
i
for all
i
jq j p
)
( ). Thus we have:
X
(
) = 21
i
i j 2I
0
(2 8)
:
.
i 2 Ig
q 2 S p
S I q x
Hence
(
)
jS I q x j Therefore
(
)
2:
i j ; j xi jj
j
q q jj x
1X
2 i j2I
1X
2 i j2I
i j
2
i j ; j xi jj
jq jjq j jj x
i j
2
i j ; j xi jj
p p jj x
(
jS I q x
Since
(
jS I q x j S I p x
= (
) for all
)
S I p x :
()
q 2 S p :
( ), hence we get, the bound (2.8).
p 2 S p
Corollary 2.7.1. If
the inequality:
i
`
2 C
(1) :=
fz 2
K
jzj
= 1 , = 1 , then we have
g
n
i
n
X
n
X
2
2
n
jjxi jj ; jj
xi jj
i=1
i=1
X
n X
n
n
X
2
2
i=1 `i i=1 `ijjxi jj ; jj i=1 `ixi jj 0:
Corollary 2.7.2. (a) If i
( = 1 2 ), then we have the inequality
2k
2k
2k
X
X
X
2
2
i 2
x
2
k
(b) If
i
x
2 H
i=1
i
jjx jj
; jj
2 H
i=1
i
i
x jj
k
jj
( = 1 2 + 1), one has:
i=1
( 1)
;
i
x jj
k
i
(2 + 1)
k
2X
k+1
i=1
2X
k+1
( 1)i
i=1
;
2
jjxi jj ; jj
2
jjxi jj + jj
2X
k+1
i=1
2X
k+1
i=1
i
2
x jj
(;1)i xi jj2 0:
0
:
SOME MAPPINGS ASSOCIATED WITH CAUCHY-BUNIAKOWSKI-SCHWARZ'S INEQUALITY 423
3. Some Properties of The Mappings (
S I J p x
) and ~(
S I p q x
)
Using the above notations, let us dene the following mappings:
:
S
f (N) Pf (N) S (R) S (K) S (H ) ! R
P
given by
(
) := (
S I J p x
and
~:
)
S I J p x
(
)
; S I p x
(
)
; S J p x
f (N) S (R) S (R) S (K) S (H ) ! R
S
P
given by
~(
) := (
S I p q x
S I p
+
q x
)
(
)
; S I p x
(
)
; S J p x
respectively.
If
= and
0, then by the inequality
f (H) are such that
(2.3), one gets that (
) 0 and if
0 then also
f (N),
(
) 0, for all
(K) and
( ).
Further on, we shall give some renements of these inequalities.
I J 2 P
I \J
S I J p x
S I p q x
I J
Then one has the inequality:
) :=
)
X
i2I
;
Proof. For all
i i i
p j j
X
j 2J
i j 2
i j
(
)
jjL I J p x jj X
j 2I
i
p jjx jj
j
j+
j
p jjx jjx
X
i2I
i i
x ;
X
j 2J
X
i2I
j
(3 1)
:
j j j
i i
i
p jjx jj
j
i
0.
p 0
p j j
N we have:
i j ; j xi jj ji j
jj x
p q be disjoint sets of positive integers and
(
(
I 2 P
x 2 S H
S I J p x
where
p 2 S
Theorem 3.1. Let
L I J p x
jjx jj ; j j jjx jj :
X
i2I
X
j 2J
i
i
p jjx jjx
j j
j jx :
i
424
S. S. DRAGOMIR AND B. MOND
By multiplying with
i j ; j xi jj 0
we derive:
jj x
2
i j ; j xi jj
jj x
(
i
j
j
i
)(
jj j j jjx jj ; j j jjx jj
i j ; j xi )jj
x
from which we deduce:
2
i j ; j xi jj
jj x
i i
j+
j
j j jjx jjx
j j
i
i
j
Now, if we multiply this inequality with
over ( )
we deduce:
i j
2 I J
(
S I J p x
j
i i
i
i
j
j j jjx jjx ; jjx jj j jx ; jjx jj j jx
)=
X
(i j )2I J
X
(i j )2I J
j
i j
p p
0 (( )
i j
2 I J
j :
) and summing
2
i j
i j ; j xi jj
i j
i i
p p jj x
j + jj jj jjxi jjxi
j
p p jj j j jjx jjx
j
i i
i
i
j j
; jjx jj j jx ; jjx jj j jx jj
(
)
jjL I J p x jj
and the inequality (3.1) is proved.
The following similar result holds for the mapping ~(
S I p q x
Theorem 3.2. With the above assumptions, for all
inequality:
~(
)
S I p q x
where
(
)=
K I p q x
X
i2I
;
i i i
p j j
X
i2 I
i i
(
)
jjK I p q x jj X
i2 I
i
q jjx jj
i
i
i+
q jjx jjx
X
i2I
X
i2I
i i i
p j jx ;
p q i i i
i2I
X
i2I
i i
(
)=
i j 2I
i j
i
p jjx jj
i j ; j xi jj
p q jj x
:
i
i
p jjx jjx
Proof. It is obvious that (see the proof of Theorem 2.1):
X
2
~
S I p q x
0 one has the
0
q j j
X
).
X
i2I
i
i i i
q j jx :
SOME MAPPINGS ASSOCIATED WITH CAUCHY-BUNIAKOWSKI-SCHWARZ'S INEQUALITY 425
Now, as in the above theorem, we have:
2
i j ; j xi jj
jj x
i i
j+
j
j j
j j jjx jjx
i
i
j
j
i i
i
i
j
j j jjx jjx ; jjx jj j jx ; jjx jj j jx
for all
N.
If we multiply this inequality with
we deduce:
j i j 2
~(
)=
S I p q x
X
(i j )2I
i j
i i
=
0(
i j 2 I
i j (i ji j
i i
i i
X
i2I
i
q jjx jj
(
i
)
j + jj jj jjxi jjxi ; j jjxj jj
i
i+
q jjx jjx
X
i2I
i j 2 I
,
2
j
jjx jjx
) and sum over
i j ; j xi jj
i j
p q jj x
j + jj jj jjxi jjxi ; j jjxj jj
j
p q
p j j
i2I
(i j )2I
i i
p q jj j j jjx jjx
X
i j )2I
(X
= i
i2I
X
;
X
i j
p q
X
i2I
i i i
p j jx ;
i i i
q j j
X
i2I
i i
X
i2I
i
p jjx jj
i
i
i2I
i
j j
i i
i
i
j
j )
j jx ; jjx jj j jx
i
p jjx jjx
X
i
j jx ; jjx jj j jx jj
i i
i q j jx
jjK I p q x jj
and the theorem is proved.
The following result also holds:
Theorem 3.3. Let
be disjoint sets of positive integers and
I J
Then one has the bound
) = sup (
(
S I J p x
where ( ) :=
S p
fp 2 S
(R)
i
q2S(p)
i
j jq j p
S I J q x
for all
i 2
p 0.
) 0
N.
g
Proof. We shall use the following identity (see the proof of Theorem 2.2):
X
2
(
)=
S I J s x
(i j )2I J
i j
i j ; j xi jj
s s jj x
:
426
S. S. DRAGOMIR AND B. MOND
Hence, for all
( ), we have:
q 2 S p
(
)
jS I J q x j X
(i j )2I J
i j
(
2
i j ; j xi jj
jq q j
jj x
)
S I J p x :
Since
( ), we get the desired bound.
p 2 S p
Similarly, we can prove
Theroem 3.4. Let
one has the bound:
~(
f (N),
I 2 P
p q 0,
2 S
~(
) = sup
(t s)2S (p)S (q)
S I p q x
(R) and
( ). Then
x 2 S H
) 0
S I t s x
:
For other recent results of Cauchy-Buniakowski-Schwarz type in inner
product spaces we refer the reader to the papers listed in the references.
References
1] S. S. Dragomir, SomerefinementsofSchwarz'sinequality, Proc. Symp. Math. Appl., (1985),
13-15.
2] S. S. Dragomir, A refinement of Cauchy-Schwarz inequality, Gaz. Mat. Metod. (Bucharest),
8 (1987), 94-95.
3] S. S. Dragomir and J. Sandor, Someinequalitiesinprehilbertianspaces, Studia Univ. BabesBolyai, Mathematica, 32:1 (1987), 71-78.
4] S. S. Dragomir, OnsomeoperatorialinequalitiesinHilbertspaces, Bull. Inst. Pol. Cluj-Napoca,
36 (1987), 23-28.
5] S. S. Dragomir, SomerefinementsofCauchy-Schwarz'sinequality, Gaz. Mat. Metod., 10
(1989), 93-95.
6] S. S. Dragomir and N. M. Ionescu, ArefinementofGram'sinequalityininnerproductspaces,
Proc. 4th Symp. Math. Appl., (1991), 188-191.
7] S. S. Dragomir and B. Mond, OnthesuperadditivityandmonotonicityofSchwarz'sinequality
ininnerproductspaces, (submitted for publication).
Department of Mathematics, Timisoara University, B-dul. V. Parvan 4, Ro-1900 Timisoara,
Romania.
School of Mathematics, La Trobe University, Bundoora, Victoria, 3083, Australia.