Table Of ContentGENEARLIZATIONS OF PRECUPANU’S INEQUALITY FOR
ORTHORNORMAL FAMILIES OF VECTORS IN INNER
5
0 PRODUCT SPACES
0
2
S.S.DRAGOMIR
n
a
J Abstract. Some genearlizations of Precupanu’s inequality for orthornormal
7 families of vectors in real or complex inner product spaces and applications
2 relatedtoBuzano’s,Richard’sandKurepa’sresultsaregiven.
]
A
F
. 1. Introduction
h
t In 1976, T. Precupanu [6] obtained the following result related to the Schwarz
a
m inequality in a real inner product space (H; , ):
h· ·i
[ Theorem 1. For any a H, x,y H 0 , we have the inequality:
∈ ∈ \{ }
1
a b + a,b x,a x,b y,a y,b x,a y,b x,y
v (1.1) −k kk k h i h ih i + h ih i 2 h ih ih i
5 2 ≤ x 2 y 2 − · x 2 y 2
k k k k k k k k
8 a b + a,b
4 k kk k h i.
≤ 2
1
0 In the right-hand side or in the left-hand side of (1.1) we have equality if and only
5 if there are λ,µ R such that
0 ∈
/ x,a y,b 1
h (1.2) λh i x+µh i y = (λa+µb).
t x 2 · y 2 · 2
a k k k k
m
Note forinstancethat[6],ify b,i.e., y,b =0,thenby(1.1)onemaydeduce:
⊥ h i
:
v a b + a,b a b + a,b
i (1.3) −k kk k h i x 2 x,a x,b k kk k h i x 2
X 2 k k ≤h ih i≤ 2 k k
r for any a,b,x H, an inequality that has been obtained previously by U. Richard
a ∈
[7]. Thecaseofequalityintheright-handsideorintheleft-handsideof(1.3)holds
if and only if there are λ,µ R with
∈
(1.4) 2λ x,a x=(λa+µb) x 2.
h i k k
For a=b, we may obtain from (1.1) the following inequality [6]
x,a 2 y,a 2 x,a y,a x,y
(1.5) 0 h i + h i 2 h ih ih i a 2.
≤ x 2 y 2 − · x 2 y 2 ≤k k
k k k k k k k k
Date:November10,2004.
1991 Mathematics Subject Classification. 46C05,46C09,26D15.
Key words and phrases. Schwarz’s inequality, Buzano’s inequality, Kurepa’s inequality,
Richard’sinequality,Precupanu’sinequality,Innerproducts.
1
2 S.S.DRAGOMIR
This inequality implies [6]:
2
x,y 1 x,a y,a 3
(1.6) h i h i + h i .
x y ≥ 2 x a y a − 2
k kk k (cid:20)k kk k k kk k(cid:21)
In [5], M.H. Moore pointed out the following reverse of the Schwarz inequality
(1.7) y,z y z , y,z H,
|h i|≤k kk k ∈
where some information about a third vector x is known:
Theorem 2. Let (H; , ) be an inner product space over the real field R and
h· ·i
x,y,z H such that:
∈
(1.8) x,y (1 ε) x y , x,z (1 ε) x z ,
|h i|≥ − k kk k |h i|≥ − k kk k
where ε is a positive real number, reasonably small. Then
(1.9) y,z max 1 ε √2ε,1 4ε,0 y z .
|h i|≥ − − − k kk k
n o
Utilising Richard’s inequality (1.3) written in the following equivalent form:
x,a x,b x,a x,b
(1.10) 2 h ih i a b a,b 2 h ih i + a b
· x 2 −k kk k≤h i≤ · x 2 k kk k
k k k k
for any a,b H and a H 0 , Precupanu has obtained the following Moore’s
∈ ∈ \{ }
type result:
Theorem 3. Let (H; , ) be a real inner product space. If a,b,x H and 0 <
h· ·i ∈
ε <ε are such that:
1 2
(1.11) ε x a x,a ε x a ,
1 2
k kk k≤h i≤ k kk k
ε x b x,b ε x b ,
1 2
k kk k≤h i≤ k kk k
then
(1.12) 2ε2 1 a b a,b 2ε2+1 a b .
1− k kk k≤h i≤ 1 k kk k
Remarkthatthe(cid:0)rightin(cid:1)equalityisalwayssa(cid:0)tisfied,s(cid:1)incebySchwarz’sinequality,
we have a,b a b . The left inequality may be useful when one assumes that
h i≤k kk k
ε (0,1]. In that case, from (1.12), we obtain
1
∈
(1.13) a b 2ε2 1 a b a,b
−k kk k≤ 1− k kk k≤h i
provided ε1 x a x,a and(cid:0)ε1 x b(cid:1) x,b , which is a refinement of
k kk k ≤ h i k kk k ≤ h i
Schwarz’s inequality
a b a,b .
−k kk k≤h i
In the complex case, apparently independent of Richard, M.L. Buzano obtained
in [2] the following inequality
a b + a,b
(1.14) x,a x,b k kk k |h i| x 2,
|h ih i|≤ 2 ·k k
provided x,a,b are vectors in the complex inner product space (H; , ).
h· ·i
Inthesamepaper[6],Precupanu,withoutmentioningBuzano’snameinrelation
to the inequality (1.14), observed that, on utilising (1.14), one may obtain the
following result of Moore type:
GENEARLIZATIONS OF PRECUPANU’S INEQUALITY 3
Theorem4. Let(H; , )bea(realor)complexinnerproductspace. Ifx,a,b H
h· ·i ∈
are such that
(1.15) x,a (1 ε) x a , x,b (1 ε) x b ,
|h i|≥ − k kk k |h i|≥ − k kk k
then
(1.16) a,b 1 4ε+2ε2 a b .
|h i|≥ − k kk k
Note that the above theoremis(cid:0)useful when, (cid:1)for ε (0,1],the quantity 1 4ε+
∈ −
2ε2 >0, i.e., ε 0,1 √2 .
∈ − 2
(cid:16) i
Remark 1. When the space is real, the inequality (1.16) provides a better lower
bound for a,b than the second bound in Moore’s result (1.9). However, it is not
|h i|
known if the firstbound in (1.9) remains valid for the case of complex spaces. From
Moore’s original proof, apparently, the fact that the space (H; , ) is real plays an
h· ·i
essential role.
Before we point out some new results for orthonormalfamilies of vectors in real
or complex inner product spaces, we state the following result that complements
the Moore type results outlined above for real spaces:
Theorem 5. Let (H; , ) be a real inner product space and a,b,x,y H 0 .
h· ·i ∈ \{ }
(i) If there exist δ ,δ (0,1] such that
1 2
∈
x,a y,a
h i δ1, h i δ2
x a ≥ y a ≥
k kk k k kk k
and δ +δ 1, then
1 2
≥
x,y 1 3
(1.17) h i (δ1+δ2)2 ( 1).
x y ≥ 2 − 2 ≥−
k kk k
(ii) If there exist µ (µ ) R such that
1 2 ∈
x,a x,b
µ a b h ih i( µ a b )
1k kk k≤ x 2 ≤ 2k kk k
k k
and 1 µ 0 ( 1 µ 0), then
≥ 1 ≥ − ≤ 2 ≤
a,b
(1.18) [ 1 ]2µ 1 h i ( 2µ +1[ 1]).
− ≤ 1− ≤ a b ≤ 2 ≤
k kk k
The proof is obvious by the inequalities (1.6) and (1.10). We omit the details.
2. Inequalities for orthonormal Families
We recall that the finite family e is orthonormal in (H; , ), a real or
{ i}i I h· ·i
complex inner product space, if ∈
0 if i=j
6
e ,e =
i j
h i
1 if i=j
where i,j I.
∈
The following result may be stated.
4 S.S.DRAGOMIR
Theorem 6. Let e and f be two finite families of orthonormal vectors
{ i}i I { j}j J
in (H; , ). For any x∈,y H 0 ∈one has the inequality
h· ·i ∈ \{ }
(2.1) x,e e ,y + x,f f ,y
i i j j
(cid:12) h ih i h ih i
(cid:12)i I j J
(cid:12)X∈ X∈
(cid:12)
(cid:12) 1 1
(cid:12)
2 x,e f ,y e ,f x,y x y .
i j i j
− h ih ih i− 2h i(cid:12)≤ 2k kk k
i I,j J (cid:12)
∈X∈ (cid:12)
The case of equality holds in (2.1) if and only if there exists a λ (cid:12)(cid:12)K such that
∈(cid:12)
(2.2) x λy =2 x,e e λ y,f f .
i i j j
− h i − h i
i I j J
X∈ X∈
Proof. We know that, if u,v H, v =0, then
∈ 6
2
u,v u 2 v 2 u,v 2
(2.3) u h i v = k k k k −|h i|
(cid:13) − v 2 · (cid:13) v 2
(cid:13) k k (cid:13) k k
(cid:13) (cid:13)
showing that, in Schw(cid:13)arz’s inequalit(cid:13)y
(cid:13) (cid:13)
(2.4) u,v 2 u 2 v 2,
|h i| ≤k k k k
the case of equality, for v =0, holds if and only if
6
u,v
(2.5) u= h i v,
v 2 ·
k k
i.e. there exists a λ R such that u=λv.
∈
Now, let u:=2 x,e e x and v :=2 y,f f y.
i Ih ii i− j Jh ji j −
Observe that ∈ ∈
P P
2
u 2 = 2 x,e e 4Re x,e e ,x + x 2
i i i i
k k (cid:13) h i (cid:13) − * h i + k k
(cid:13)(cid:13) Xi∈I (cid:13)(cid:13) Xi∈I
=(cid:13)4 x,e 2 (cid:13)4 x,e 2+ x 2 = x 2,
(cid:13) |h ii| −(cid:13) |h ii| k k k k
i I i I
X∈ X∈
and, similarly
v 2 = y 2.
k k k k
Also,
u,v =4 x,e f ,y e ,f + x,y
i j i j
h i h ih ih i h i
i I,j J
∈X∈
2 x,e e ,y 2 x,f f ,y .
i i j j
− h ih i− h ih i
i I j J
X∈ X∈
Therefore,bySchwarz’sinequality(2.4)wededuce the desiredinequality(2.1). By
(2.5),thecaseofequalityholdsin(2.1)ifandonlyifthereexistsaλ Ksuchthat
∈
2 x,e e x=λ 2 y,f f y ,
i i j j
h i − h i −
i I j J
X∈ X∈
which is equivalent to (2.2).
GENEARLIZATIONS OF PRECUPANU’S INEQUALITY 5
Remark 2. If in (2.2) we choose x=y, then we get the inequality:
1
(2.6) x,e 2+ x,f 2 2 x,e f ,x e ,f x 2
i j i j i j
(cid:12) |h i| |h i| − h ih ih i− 2k k (cid:12)
(cid:12)i I j J i I,j J (cid:12)
(cid:12)X∈ X∈ ∈X∈ (cid:12)
(cid:12) 1(cid:12)
(cid:12) (cid:12)x 2
(cid:12) (cid:12)
≤ 2k k
for any x H.
∈
If in the above theorem we assume that I = J and f = e , i I, then we get
i i
∈
from (2.1) the Schwarz inequality x,y x y .
If I J = ∅, I J = K, g |=h e i,|k≤k Ik,kgk = f , k J and g is
∩ ∪ k k ∈ k k ∈ { k}k K
orthonormal, then from (2.1) we get: ∈
1 1
(2.7) x,g g ,y x,y x y , x,y H
k k
(cid:12) h ih i− 2h i(cid:12)≤ 2k kk k ∈
(cid:12)(cid:12)kX∈K (cid:12)(cid:12)
which has bee(cid:12)n obtained earlier by the aut(cid:12)hor in [3].
(cid:12) (cid:12)
If I andJ reduce to one element, namely e = e , f = f with e,f =0, then
1 e 1 f 6
k k k k
from (2.1) we get
x,e e,y x,f f,y x,e f,y e,f 1
(2.8) h ih i + h ih i 2 h ih ih i x,y
(cid:12) e 2 f 2 − · e 2 f 2 − 2h i(cid:12)
(cid:12) k k k k k k k k (cid:12)
(cid:12) 1 (cid:12)
(cid:12) x y ,(cid:12) x,y H
(cid:12) (cid:12)
≤ 2k kk k ∈
which is the corresponding complex version of Precupanu’s inequality (1.1).
If in (2.8) we assume that x=y, then we get
x,e 2 x,f 2 x,e f,e e,f 1 1
(2.9) |h i| + |h i| 2 h ih ih i x 2 x 2.
(cid:12) e 2 f 2 − · e 2 f 2 − 2k k (cid:12)≤ 2k k
(cid:12) k k k k k k k k (cid:12)
(cid:12) (cid:12)
The fol(cid:12)lowing corollary may be stated: (cid:12)
(cid:12) (cid:12)
Corollary 1. With the assumptions of Theorem 6, we have:
(2.10) x,e e ,y + x,f f ,y 2 x,e f ,y e ,f
i i j j i j i j
(cid:12) h ih i h ih i − h ih ih i(cid:12)
(cid:12)i I j J i I,j J (cid:12)
(cid:12)X∈ X∈ ∈X∈ (cid:12)
(cid:12) (cid:12)
(cid:12) 1 (cid:12)
(cid:12) (cid:12)
x,y + x,e e ,y + x,f f ,y
i i j j
≤ 2|h i| (cid:12) h ih i h ih i
(cid:12)i I j J
(cid:12)X∈ X∈
(cid:12)
(cid:12) 1
(cid:12)
2 x,e f ,y e ,f x,y
i j i j
− h ih ih i− 2|h i|(cid:12)
i I,j J (cid:12)
∈X∈ (cid:12)
1 (cid:12)
[ x,y + x y ]. (cid:12)
≤ 2 |h i| k kk k (cid:12)
Proof. The first inequality follows by the triangle inequality for the modulus. The
secondinequalityfollowsby (2.1)onaddingthe quantity 1 x,y onbothsides.
2|h i|
6 S.S.DRAGOMIR
Remark 3. (1) If we choose in (2.10), x=y, then we get:
(2.11) x,e 2+ x,f 2 2 x,e f ,x e ,f
i j i j i j
(cid:12) |h i| |h i| − h ih ih i(cid:12)
(cid:12)i I j J i I,j J (cid:12)
(cid:12)X∈ X∈ ∈X∈ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) x,e 2+ x,f 2 (cid:12)
i j
≤(cid:12) |h i| |h i|
(cid:12)i I j J
(cid:12)X∈ X∈
(cid:12)
(cid:12) 1 1
(cid:12) 2 x,e f ,x e ,f x 2 + x 2
i j i j
− h ih ih i− 2k k (cid:12) 2k k
i I,j J (cid:12)
∈X∈ (cid:12)
x 2. (cid:12)(cid:12)
≤k k (cid:12)
We observe that (2.11) will generate Bessel’s inequality if e , f
{ i}i I { j}j J
are disjoint parts of a larger orthonormal family. ∈ ∈
(2) From (2.8) one can obtain:
x,e e,y x,f f,y x,e f,y e,f 1
(2.12) h ih i + h ih i 2 h ih ih i [ x y + x,y ]
(cid:12) e 2 f 2 − · e 2 f 2 (cid:12)≤ 2 k kk k |h i|
(cid:12) k k k k k k k k (cid:12)
(cid:12) (cid:12)
(cid:12)and in particular (cid:12)
(cid:12) (cid:12)
x,e 2 x,f 2 x,e f,e e,f
(2.13) |h i| + |h i| 2 h ih ih i x 2,
(cid:12) e 2 f 2 − · e 2 f 2 (cid:12)≤k k
(cid:12) k k k k k k k k (cid:12)
(cid:12) (cid:12)
for any(cid:12)x,y H. (cid:12)
(cid:12) ∈ (cid:12)
The case of real inner products will provide a natural genearlization for Precu-
panu’s inequality (1.1):
Corollary 2. Let (H; , ) be a real inner product space and e , f two
h· ·i { i}i I { j}j J
finite families of orthonormal vectors in (H; , ). For any x,y ∈H 0 o∈ne has
h· ·i ∈ \{ }
the double inequality:
1
(2.14) [ x,y x y ]
2 |h i|−k kk k
x,e y,e + x,f y,f 2 x,e y,f e ,f
i i j j i j i j
≤ h ih i h ih i− h ih ih i
i I j J i I,j J
X∈ X∈ ∈X∈
1
[ x y + x,y ].
≤ 2 k kk k |h i|
In particular, we have
(2.15) 0 x,e 2+ x,f 2 2 x,e x,f e ,f
i j i j i j
≤ h i h i − h ih ih i
i I j J i I,j J
X∈ X∈ ∈X∈
x 2,
≤k k
for any x H.
∈
Remark 4. Similar particular inequalities to those incorporated in (2.7) – (2.13)
may be stated, but we omit them.
GENEARLIZATIONS OF PRECUPANU’S INEQUALITY 7
3. Refinements of Kurepa’s Inequality
Let (H; , ) be a real inner product space generating the norm . The com-
h· ·i k·k
plexification HC of H is defined as a complex linear space H H of all ordered
×
pairs (x,y), x,y H endowed with the operations:
∈
(x,y)+(x′,y′):=(x+x′,y+y′), x,x′,y,y′ H;
∈
(σ+iτ) (x,y):=(σx τy,τx+σy), x,y H and σ,τ R.
· − ∈ ∈
On HC := H H, endowed with the above operations, one can now canonically
×
define the scalar product , by:
C
h· ·i
(3.1) z,z := x,x + y,y +i[ x,y x,y ]
′ C ′ ′ ′ ′
h i h i h i h i−h i
where z =(x,y), z′ =(x′,y′) HC. Obviously,
∈
z 2C = x 2+ y 2, z =(x,y) HC.
k k k k k k ∈
Onecanalsodefinetheconjugate ofavectorz =(x,y)byz¯:=(x, y).Itiseasyto
−
seethat,theelementsofHC,underdefinedoperations,behaveasformal“complex”
combinationsx+iy withx,y H.Becauseofthis,wemaywritez =x+iy instead
∈
of z =(x,y). Thus, z¯=x iy.
−
Under this setting, S. Kurepa [4] proved the following refinement of Schwarz’s
inequality:
1
(3.2) a,z 2 a 2 z 2 + z,z¯ a 2 z 2 ,
C C C C
|h i | ≤ 2k k k k |h i | ≤k k k k
h i
for any a H and z HC.
∈ ∈
ThiswasmotivatedbygeneralisingthedeBruijnresultforsequencesofrealand
complex numbers obtained in [1].
The following result may be stated.
Theorem 7. Let (H; , ) be a real inner product space and e , f two
h· ·i { i}i I { j}j J
finitefamilies in H.If (HC; , C) is thecomplexification of (H; , ∈),then f∈or any
h· ·i h· ·i
w HC, we have the inequalities
∈
(3.3) w,e 2 + w,f 2 2 w,e w,f e ,f
(cid:12) h iiC h jiC− h iiCh jiCh i ji(cid:12)
(cid:12)i I j J i I,j J (cid:12)
(cid:12)X∈ X∈ ∈X∈ (cid:12)
(cid:12) (cid:12)
(cid:12) 1 (cid:12)
(cid:12) w,w¯ + w,e 2 + w,f 2 (cid:12)
≤ 2|h iC| (cid:12) h iiC h jiC
(cid:12)i I j J
(cid:12)X∈ X∈
(cid:12)
(cid:12) 1
(cid:12)
2 w,e w,f e ,f w,w¯
− h iiCh jiCh i ji− 2h iC(cid:12)
i I,j J (cid:12)
∈X∈ (cid:12)
1 (cid:12)
w 2 + w,w¯ w 2 . (cid:12)
C C C (cid:12)
≤ 2 k k |h i | ≤k k
h i
Proof. Define gj HC, gj :=(ej,0), j I. For any k,j I we have
∈ ∈ ∈
g ,g = (e ,0),(e ,0) = e ,e =δ ,
k j C k j C k j kj
h i h i h i
therefore {gj}j I is an orthonormal family in (HC;h·,·iC).
∈
8 S.S.DRAGOMIR
If we apply Corollary 1 for (HC; , C), x=w, y =w¯, we may write:
h· ·i
(3.4) w,e e ,w¯ + w,f f ,w¯
(cid:12) h iiCh i iC h jih j i
(cid:12)i I j J
(cid:12)X∈ X∈
(cid:12)
(cid:12)
(cid:12)
2 w,e f ,w e ,f
− h iiCh j iCh i ji(cid:12)
i I,j J (cid:12)
∈X∈ (cid:12)
(cid:12)
1 (cid:12)
(cid:12)
w w¯ + w,e e ,w¯ + w,f f ,w¯
≤ 2k kCk kC (cid:12) h iiCh i iC h jih j i
(cid:12)i I j J
(cid:12)X∈ X∈
(cid:12)
(cid:12) 1
(cid:12)
2 w,e f ,w e ,f w,w¯
− h iiCh j iCh i ji− 2h iC(cid:12)
i I,j J (cid:12)
∈X∈ (cid:12)
1 (cid:12)
[ w,w¯ + w w¯ ]. (cid:12)
≤ 2 |h iC| k kCk kC (cid:12)
However,for w :=(x,y) HC, we have w¯ =(x, y) and
∈ −
e ,w¯ = (e ,0),(x, y) = e ,x +i e ,y
j C j C j j
h i h − i h i h i
and
w,e = (x,y),(e ,0) = x,e +i e ,y
j C j C j j
h i h i h i h i
showing that e ,w¯ = w,e for any j I. A similar relation is true for f and
j j j
h i h i ∈
since
1
w = w¯ = x 2+ y 2 2 ,
C C
k k k k k k k k
hence from (3.4) we deduce the desired in(cid:16)equality (3.3(cid:17)).
Remark 5. It is obvious that, if one family, say f is empty, then, on ob-
{ j}j J
serving that all sums should be zero, from (3.3) on∈e would get [3]
j J
∈
P
(3.5) w,e 2
i C
(cid:12) h i (cid:12)
(cid:12)(cid:12)Xi∈I (cid:12)(cid:12)
(cid:12) 1 (cid:12) 1
(cid:12) w,w¯ (cid:12) + w,e 2 w,w¯
C i C C
≤ 2|h i | (cid:12) h i − 2h i (cid:12)
1 (cid:12)(cid:12)Xi∈I (cid:12)(cid:12)
w 2C+ w(cid:12)(cid:12),w¯ C w 2C. (cid:12)(cid:12)
≤ 2 k k |h i | ≤k k
h i
If in (3.5) one assumes that the family e contains only one element e =
a ,a=0, then by selecting w =z, one wo{ulid}id∈eIduce
a 6
k k
1 1
a,z 2 a,z 2 z,z¯ + z,z¯
C C C C
|h i | ≤ h i − 2h i 2|h i |
(cid:12) (cid:12)
(cid:12)1 (cid:12)
(cid:12) a 2 z 2 + z,(cid:12)z¯ ,
(cid:12) C (cid:12) C
≤ 2k k k k |h i |
h i
which is a refinement for Kurepa’s inequality (3.2).
Acknowledgement 1. The author would like to thank the anonymous referee for
his/her comments that have been implemented in the final version of this paper.
GENEARLIZATIONS OF PRECUPANU’S INEQUALITY 9
References
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School of Computer Scienceand Mathematics,Victoria University of Technology,
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E-mail address: [email protected]
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