INEQUALITIES INVOLVING THE INNER PRODUCT Communicated by S.S. Dragomir

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Volume 8 (2007), Issue 3, Article 86, 4 pp.
INEQUALITIES INVOLVING THE INNER PRODUCT
DORIAN POPA AND IOAN RAŞA
T ECHNICAL U NIVERSITY OF C LUJ -NAPOCA
D EPARTMENT OF M ATHEMATICS
S TR . C. DAICOVICIU 15
C LUJ -NAPOCA , ROMANIA
[email protected]
[email protected]
Received 20 February, 2007; accepted 16 July, 2007
Communicated by S.S. Dragomir
A BSTRACT. The paper contains inequalities related to generalizations of Schwarz’s inequality.
Key words and phrases: Inner product, Schwarz inequality.
2000 Mathematics Subject Classification. 26D20.
1. I NTRODUCTION
Let (H, h·, ·i) be a Hilbert space over the field K = R or K = C. Then for all x, a, b ∈ H
the following inequality holds:
1
2
ha, xihx, bi − ha, bikxk ≤ 1 kak kbk kxk2 .
(1.1)
2
2
In particular, for a = b (1.1) reduces to Schwarz’s inequality.
For historical remarks, proofs, extensions, generalizations and applications of (1.1), see [1] –
[5] and the references given therein.
In this paper we consider a suitable quadratic form and derive inequalities related to (1.1).
More precisely, we obtain estimates involving the real and the imaginary part of the expression
whose absolute value is contained in the left hand of (1.1).
2. T HE R ESULTS
Let v1 , . . . , vn (n ≥ 2) be linearly independent vectors in H, and v := v1 + · · · + vn .
061-07
2
D ORIAN P OPA AND I OAN R A ŞA
Consider the matrix

hvn , v − v1 i

 hv , v − v i hv , v − v i · · · hv , v − v i
 1
2
2
2
n
2
A := 
.
.
.

..
..
..
···

hv1 , v − vn i hv2 , v − vn i · · · hvn , v − vn i





hv1 , v − v1 i hv2 , v − v1 i · · ·
Theorem 2.1.
(i) The matrix A has real eigenvalues λ1 ≤ · · · ≤ λn ; moreover, λ1 ≤ 0 and λn ≥ 0.
(ii) The following inequalities hold:
(2.1)
2
λ1 kxk ≤
n
X
hvi , xihx, vj i ≤ λn kxk2 ,
x ∈ H.
i,j=1
i6=j
Proof. Let Hn denote the linear subspace of H generated by v1 , . . . , vn . Consider the linear
operator T : Hn → Hn defined by
T x = hx, viv −
n
X
hx, vi ivi ,
x ∈ Hn .
i=1
Then for all x, y ∈ Hn we have
hT x, yi = hx, vihv, yi −
= hy, vihx, vi −
*
=
x, hy, viv −
n
X
hx, vi ihvi , yi
i=1
n
X
i=1
n
X
hy, vi ihx, vi i
+
hy, vi ivi
i=1
= hx, T yi.
We conclude that T is self-adjoint, hence it has real eigenvalues λ1 ≤ · · · ≤ λn and:
(2.2)
λ1 kxk2 ≤ hT x, xi ≤ λn kxk2 ,
x ∈ Hn .
On the other hand,
T vj = hvj , viv −
n
X
hvj , vi ivi
i=1
=
n
X
(hvj , vi − hvj , vi i)vi =
i=1
n
X
hvj , v − vi ivi
i=1
for all j = 1, 2, . . . , n.
This means that A is the matrix of T with respect to the basis {v1 , v2 , . . . , vn } of Hn , and so
λ1 ≤ · · · ≤ λn are the eigenvalues of the matrix A.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 86, 4 pp.
http://jipam.vu.edu.au/
I NEQUALITIES I NVOLVING THE I NNER P RODUCT
3
Now remark that
hT x, xi = hx, vihv, xi −
n
X
hx, vi ihvi , xi
i=1
=
=
n
X
hvi , xi
i=1
n
X
n
X
hx, vi i −
i=1
hvi , xihx, vj i,
n
X
hvi , xihx, vi i
i=1
x ∈ Hn .
i,j=1
i6=j
Combined with (2.2), this gives
(2.3)
2
λ1 kxk ≤
n
X
hvi , xihx, vj i ≤ λn kxk2 ,
x ∈ Hn .
i,j=1
i6=j
Let x ∈ Hn , x 6= 0, hx, vi i = 0, i = 1, 2, . . . , n − 1.
From (2.3) we infer that
λ1 ≤ 0 ≤ λn .
(2.4)
Let y ∈ H. Then y = x + z, x ∈ Hn , z ∈ Hn⊥ and kyk2 = kxk2 + kzk2 , so that kyk2 ≥ kxk2 .
Moreover,
hvi , yi = hvi , x + zi = hvi , xi, i = 1, . . . , n.
Using (2.3) and (2.4), we get
2
2
λ1 kyk ≤ λ1 kxk ≤
n
X
hvi , yihy, vj i ≤ λn kxk2 ≤ λn kyk2
i,j=1
i6=j
and this concludes the proof.
Corollary 2.2. Let a, b, x ∈ H. Then
p
1
1
2
≤ kxk2 kak2 kbk2 − (Imha, bi)2 .
(2.5)
Re
ha,
xihx,
bi
−
kxk
ha,
bi
2
2
Proof. If a and b are linearly dependent, (2.5) can be verified directly. Otherwise it is a consequence of Theorem 2.1.
Indeed for n = 2, v1 = a, v2 = b, the eigenvalues of the matrix A are
p
λ1,2 = Reha, bi ± kak2 kbk2 − (Imha, bi)2
and
hT x, xi = 2 Rehx, aihb, xi,
x ∈ H.
Remark 2.3.
(i) When K = R, (2.5) coincides with (1.1).
(ii) Let K = C. Applying Corollary 2.2 to the vectors ia, b, x we get
p
1
1
2
≤ kxk2 kak2 kbk2 − (Reha, bi)2 .
(2.6)
Im
ha,
xihx,
bi
−
kxk
ha,
bi
2
2
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 86, 4 pp.
http://jipam.vu.edu.au/
4
D ORIAN P OPA AND I OAN R A ŞA
R EFERENCES
[1] M.L. BUZANO, Generalizzazione della disuguglianza di Cauchy-Schwarz (Italian), Rend. Sem.
Mat. Univ. e Politech. Torino, 31(1971/73), 405–409(1974).
[2] S.S. DRAGOMIR, Refinements of Buzano’s and Kurepa’s inequalities in inner product spaces, Facta
Univ. (Niš), Ser. Math. Inform., 20 (2005), 65–73.
[3] S.S. DRAGOMIR, A potpourri of Schwarz related inequalities in inner product spaces (I), J. Inequal.
Pure Appl. Math., 6(3) (2005), Art. 59. [ONLINE: http://jipam.vu.edu.au/article.
php?sid=532].
[4] J.E. PEČARIĆ, On some classical inequalities in unitary spaces, Mat. Bilten, 16 (1992), 63–72.
[5] T. PRECUPANU, On a generalization of Cauchy-Buniakowski-Schwarz inequality, Anal. St. Univ.
"Al. I. Cuza" Iaşi, 22(2) (1976), 173–175.
J. Inequal. Pure and Appl. Math., 8(3) (2007), Art. 86, 4 pp.
http://jipam.vu.edu.au/
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