INEQUALITIES INVOLVING THE INNER PRODUCT

(1)

Inequalities Involving the Inner Product Dorian Popa and Ioan Ra¸sa vol. 8, iss. 3, art. 86, 2007

Title Page

Contents

JJ II

J I

Page1of 8 Go Back Full Screen

INEQUALITIES INVOLVING THE INNER PRODUCT

DORIAN POPA AND IOAN RA ¸SA

Technical University of Cluj-Napoca Department of Mathematics Str. C. Daicoviciu 15 Cluj-Napoca, Romania

EMail:{Popa.Dorian,Ioan.Rasa}@math.utcluj.ro

Received: 20 February, 2007

Accepted: 16 July, 2007

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 26D20.

Key words: Inner product, Schwarz inequality.

Abstract: The paper contains inequalities related to generalizations of Schwarz’s inequality.

(2)

Title Page Contents

JJ II

J I

1. Introduction

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:

(1.1)

ha, xihx, bi − 1

2ha, bikxk²

≤ 1

2kak kbk kxk². In particular, fora=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).

(4)

Title Page Contents

JJ II

J I

Close

2. The Results

Letv₁, . . . , v_n(n≥2)be linearly independent vectors inH, andv :=v₁+· · ·+v_n. Consider the matrix

A:=







hv1, v−v1i hv2, v−v1i · · · hvn, v−v1i hv1, v−v2i hv2, v−v2i · · · hvn, v−v2i

... ... · · · ... hv₁, v−v_ni hv₂, v−v_ni · · · hv_n, v−v_ni







Theorem 2.1.

(i) The matrix A has real eigenvalues λ₁ ≤ · · · ≤ λ_n; moreover, λ₁ ≤ 0 and λ_n ≥0.

(ii) The following inequalities hold:

(2.1) λ1kxk² ≤

n

X

i,j=1 i6=j

hvi, xihx, vji ≤λnkxk², x∈H.

Proof. LetHn denote the linear subspace of H generated by v1, . . . , vn. Consider the linear operatorT :Hn→Hndefined by

T x=hx, viv−

n

X

i=1

hx, v_iiv_i, x∈H_n.

Then for allx, y ∈H_nwe have

hT x, yi=hx, vihv, yi −

n

Xhx, viihvi, yi

(5)

Title Page Contents

JJ II

J I

Close

=hy, vihx, vi −

n

X

i=1

hy, v_iihx, v_ii

=

*

x,hy, viv−

n

X

i=1

hy, v_iiv_i +

=hx, T yi.

We conclude thatT is self-adjoint, hence it has real eigenvaluesλ₁ ≤ · · · ≤λ_nand:

(2.2) λ₁kxk² ≤ hT x, xi ≤λ_nkxk², x∈H_n. On the other hand,

T v_j =hv_j, viv−

n

X

i=1

hv_j, v_iiv_i

=

n

X

i=1

(hv_j, vi − hv_j, v_ii)v_i =

n

X

i=1

hv_j, v−v_iiv_i for allj = 1,2, . . . , n.

This means thatAis the matrix ofT with respect to the basis{v₁, v₂, . . . , v_n}of H_n, and soλ₁ ≤ · · · ≤λ_nare the eigenvalues of the matrixA.

Now remark that

hT x, xi=hx, vihv, xi −

n

X

i=1

hx, v_iihv_i, xi

=

n

Xhv_i, xi

n

Xhx, v_ii −

n

Xhv_i, xihx, v_ii

(6)

Title Page Contents

JJ II

J I

Close

=

n

X

i,j=1 i6=j

hv_i, xihx, v_ji, x∈H_n.

Combined with (2.2), this gives (2.3) λ1kxk² ≤

n

X

i,j=1 i6=j

hvi, xihx, vji ≤λnkxk², x∈Hn.

Letx∈H_n,x6= 0,hx, v_ii= 0,i= 1,2, . . . , n−1.

From (2.3) we infer that

(2.4) λ₁ ≤0≤λ_n.

Lety ∈ H. Theny = x+z, x ∈ Hn, z ∈ H_n^⊥ andkyk² = kxk² +kzk², so that kyk² ≥ kxk². Moreover,

hv_i, yi=hv_i, x+zi=hv_i, xi, i= 1, . . . , n.

Using (2.3) and (2.4), we get λ₁kyk² ≤λ₁kxk² ≤

n

X

i,j=1 i6=j

hv_i, yihy, v_ji ≤λ_nkxk² ≤λ_nkyk²

and this concludes the proof.

Corollary 2.2. Leta, b, x ∈H. Then (2.5)

Re

ha, xihx, bi − 1

kxk²ha, bi

≤ 1

kxk²p

kak²kbk²−(Imha, bi)².

(7)

Title Page Contents

JJ II

J I

Proof. Ifa andbare linearly dependent, (2.5) can be verified directly. Otherwise it is a consequence of Theorem2.1.

Indeed forn= 2,v₁ =a,v₂ =b, the eigenvalues of the matrixAare λ_1,2 = Reha, bi ±p

kak²kbk²−(Imha, bi)² and

hT x, xi= 2 Rehx, aihb, xi, x∈H.

Remark 1.

(i) WhenK =R, (2.5) coincides with (1.1).

(ii) LetK =C. Applying Corollary2.2to the vectorsia,b,xwe get (2.6)

Im

ha, xihx, bi −1

2kxk²ha, bi

≤ 1

2kxk²p

kak²kbk²−(Reha, bi)².

(8)

Title Page Contents

JJ II

J I

References

[1] M.L. BUZANO, Generalizzazione della disuguglianza di Cauchy-Schwarz (Ital- ian), 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. PE ˇCARI ´C, On some classical inequalities in unitary spaces, Mat. Bilten, 16 (1992), 63–72.

[5] T. PRECUPANU, On a generalization of Cauchy-Buniakowski-Schwarz in- equality, Anal. St. Univ. "Al. I. Cuza" Ia¸si, 22(2) (1976), 173–175.