Some Inequalities in 2-inner Product Spaces

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Some Inequalities in 2-inner Product Spaces

This is the Published version of the following publication

Cho, Yeol Je, Dragomir, Sever S, White, A and Kim, Seong Sik (1999) Some Inequalities in 2-inner Product Spaces. RGMIA research report collection, 2 (2).

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Y.J. CHO, S.S. DRAGOMIR, A. WHITE, AND S.S. KIM

Abstract. In this paper we extend some results on the refinement of Cauchy- Buniakowski-Schwarz’s inequality and A´czel’s inequality in inner product spaces to 2−inner product spaces.

1. Introduction

Let X be a real linear space of dimension greater than 1 and let k·,·k be a real-valued function onX×X satisfying the following conditions:

(N1) kx, yk= 0 if and only ifxandy are linearly dependent;

(N2) kx, yk=ky, xk;

(N₃) kαx, yk=|α| kx, yk for any real numberα;

(N4) kx, y+zk ≤ kx, yk+kx, zk.

k·,·kis called a 2−norm onX and (X,k·,·k)a linear 2–normed space cf. [10].

Some of the basic properties of the 2-norms are that they are nonnegative, and kx, y+αxk=kx, ykfor every x, y inX and every real numberα.

For any non-zerox₁, x₂, ..., x_n inX, letV (x₁, x₂, ..., x_n) denote the subspace of X generated by x₁, x₂, ..., x_n. Whenever the notationV (x₁, x₂, ..., x_n) is used, we will understand thatx₁, x₂, ..., x_n are linearly independent.

A concept which is closely related to linear 2-normed space is that of 2 inner product spaces. For a linear spaceX of dimension greater than 1 let (·,· | ·) be a real-valued function onX×X×X which satisfies the following conditions:

(I1) (x, x|z)≥0; (x, x|z) = 0 if and only ifxandzare linearly dependent;

(I2) (x, x|z) = (z, z|x) ; (I3) (x, y|z) = (y, x|z) ;

(I4) (αx, y|z) =α(x, y |z) for any real numberα;

(I₅) (x+x⁰, y|z) = (x, y|z) + (x⁰, y|z).

(·,· | ·) is called a 2-innerproduct and (X,(·,· | ·)) a 2-inner product space([3]).

These spaces are studied extensively in [1], [2], [4]-[6] and [11]. In [3] it is shown that kx, zk = (x, x|z)¹² is a 2−norm on (X,k·,·k). Every 2−inner product space will be considered to be a linear 2−normed space with the 2−norm

1991Mathematics Subject Classification. Primary 26D99; Secondary 46Cxx.

Key words and phrases. 2-Inner Product, Cauchy-Buniakowski-Schwarz’s Inequality, Triangle Inequality, A´czel’s Inequality.

This paper has been supported in part by NON-DIRECTED RESEARCH FUND, Korea Research Foundation, 1996.

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2 Y.J. CHO, S.S. DRAGOMIR, A. WHITE, AND S.S. KIM

kx, zk = (x, x|z)¹². R. Ehret, [9], has shown that for any 2−inner product space (X,(·,· | ·)),kx, zk= (x, x|z)¹² defines a 2-norm for which

(x, y|z) =1 4

kx+y, zk²− kx−y, zk²

, (1.1)

kx+y, zk²+kx−y, zk²= 2

kx, zk²+ky, zk²

. (1.2)

Besides, if (X,k·,·k) is a linear 2−normed space in which condition (1.2), being a 2-dimensional analogue of the parallelogram law, is satisfied for everyx, y, z∈X, then a 2-inner product onX is defined on by (1.1).

For a 2-inner product space (X,(·,· | ·)) Cauchy-Schwarz’s inequality

|(x, y|z)| ≤(x, x|z)¹²(y, y|z)¹² =kx, zk ky, zk,

a 2−dimensional analogue of Cauchy-Buniakowski-Schwarz’s inequality, holds (cf.

[3]).

2. Refinements of Cauchy-Schwarz’s Inequality

Throughout this paper, let (X,(·,· | ·)) denote a 2−inner product space with kx, zk= (x, x|z)¹²,Rthe set of real numbers andNthe set of natural numbers.

Theorem 2.1. Let x, y, z, u, v∈X withz /∈V (x, y, u, v)be such that ku, zk²≤2 (x, u|z), kv, zk²≤2 (y, v|z). (2.1)

Then, we have the inequality

2 (x, u|z)− ku, zk²¹₂

2 (y, v|z)− kv, zk²¹₂ (2.2)

+|(x, y|z)−(x, v|z)−(u, y|z) + (u, v|z)| ≤ kx, zk ky, zk. Proof. Note that

m²−n²

p²−q²

≤(mp−nq)² (2.3)

for everym, n, p, q∈R.Since

|(x, y |z)−(x, v |z)−(u, y|z) + (u, v|z)|²

=|(x−u, y−v|z)|²≤ kx−u, zk²ky−v, zk²

=

kx, zk²+ku, zk²−2 (x, u|z) ky, zk²+kv, zk²−2 (y, v|z) , by (2.3), we have

|(x, y |z)−(x, v |z)−(u, y|z) + (u, v|z)|² (2.4)

≤

kx, zk ky, zk −

2 (x, u|z)− ku, zk² ¹₂

2 (y, v|z)− kv, zk² ¹₂²

. On the other hand

0≤

2 (x, u|z)− ku, zk² ¹₂

≤ kx, zk, 0≤

2 (y, v|z)− kv, zk²¹₂

≤ ky, zk,

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which imply

2 (x, u|z)− ku, zk²¹₂

2 (y, v|z)− kv, zk²¹₂

≤ kx, zk ky, zk. Therefore, from (2.4), we have the inequality (2.2).This completes the proof.

Corollary 2.2. Letx, y, z, e∈X be such thatke, zk= 1andz /∈V (x, y, e).Then

|(x, y|z)| ≤ |(x, y|z)−(x, e|z) (e, y|z)| (2.5)

+|(x, e|z) (e, u|z)| ≤ kx, zk ky, zk.

Proof. If we putu= (x, e|z)eandv= (y, e|z)e,then the conditions (2.1) hold.

In fact,

2 (x, u|z)− ku, zk²= 2 (x,(x, e|z)e|z)− k(x, e|z)e, zk²

= 2 (x, e|z) (x, e|z)−(x, e|z)²= (x, e|z) (x, e|z)≥0.

And similarly for the second condition in (2.1). Moreover,

|(x, y|z)−(x, v|z)−(u, y|z) + (u, v|z)|

=|(x, y |z)−(x, e|z) (y, e|z)−(x, e|z) (e, y|z) + (x, e|z) (y, e|z)|

=|(x, y |z)−(x, e|z) (e, y|z)| so, by Theorem 2.1, we have (2.5).

Corollary 2.3. Let x, y, z ∈ X be such that kx, zk² ≤ 2,ky, zk² ≤ 2 and z /∈ V (x, y). Then

|(x, y|z)|²

2− kx, zk²¹₂

2− ky, zk²¹₂ (2.6)

+|(x, y |z)|

1− kx, zk²− ky, zk²+ (x, y|z)

2

≤ kx, zk ky, zk.

Proof. If we putu= (x, y|z)yandv= (y, x|z)x,then the inequality (2.3) holds.

Moreover, we have

2 (x, u|z)− ku, zk²¹₂

2 (y, v|z)− kv, zk²¹₂

=|(x, y|z)|²

2− kx, zk²¹₂

2− ky, zk²¹₂ ,

|(x, y|z)−(x, v|z)−(u, y|z) + (u, v|z)|

=|(x, y|z)|

1− kx, zk²− ky, zk²+|(x, y|z)|² . Therefore, by Theorem 2.1, we have the inequality (2.6).

Theorem 2.4. Let x, y, z, e∈X be such thatke, zk= 1 andz /∈V(x, y, e).Then

|(x, y|z)−(x, e|z) (e, y|z)|² (2.7)

≤

kx, zk²− |(x, e|z)|² ky, zk²− |(y, e|z)|²

.

Proof. Consider a mappingP :X×X×X →Rdefined byP(x, y, z) = (x, y|z)− (x, e|z) (e, y|z) for everyx, y, z, e∈X,having the properties:

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(i) P(x, x, z)≥0,

(ii) P(αx+βx⁰, y, z) =P(x, y, z) +βP(x⁰, y, z), (iii) P(x, y, z) =P(y, x, z).

Then Cauchy-Schwarz’s inequality

|P(x, y, z)|²≤P(x, x, z)P(y, y, z) (2.8)

holds.

Indeed, we observe that

0≤P(x+αP(x, y, z)y, x+αP(x, y, z)y, z)

=P(x, x, z) + 2αP(x, y, z)²+α²P(x, y, z)²P(y, y, z) (∀)α∈R.

It is well known that ifa≥0 and

aα²+βα+c≥0 for allα∈R, then ∆ =b²−4ac≤0.

Then by the above inequality we deduce

P(x, y, z)⁴≤P(x, x, z)P(y, y, z)P(x, y, z)². (2.9)

IfP(x, y, z) = 0 then (2.8) holds.

IfP(x, y, z)6= 0 then we can devide in (2.9) byP(x, y, z) and obtain (2.8). The theorem is thus proved.

Remark 2.1. By the inequalities (2.3)and(2.7),we have

|(x, y|z)−(x, e|z) (e, y|z)|²

≤

kx, zk²− |(x, e|z)|² ky, zk²− |(y, e|z)|²

≤(kx, zk ky, zk − |(x, e|z) (e, y|z)|)². Sincekx, zk ky, zk ≥ |(x, e|z) (e, y|z)|, we get

|(x, y|z)−(x, e|z) (e, y|z)| ≤ kx, zk ky, zk − |(x, e|z) (e, y|z)|, which yields the inequality (2.5).

Corollary 2.5. Letx, y, z, e∈X be such thatke, zk= 1andz /∈V (x, y, e).Then

kx+y, zk²− |(x+y, e|z)|²¹₂ (2.10)

≤

kx, zk²− |(x, e|z)|²¹₂ +

ky, zk²− |(y, e|z)|²¹₂ .

Proof. If we defineS : X×X →R by S(x, z) =P(x, x, z)¹² for every x, y ∈X and use the triangle inequality forS(x, z),then we have (2.10).

Corollary 2.6. For every non-zerox, y, z, u∈X,with z /∈V(x, y, u), we have

(x, y|z) kx, zk ky, zk

2

+

(y, u|z) ky, zk ku, zk

2

+

(u, x|z) ku, zk kx, zk

2

(2.11)

≤1 + 2

(x, y|z) (y, u|z) (u, x|z) kx, zk²ky, zk²ku, zk²

.

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For the proof of next theorem, we need the following lemma:

Lemma 2.7. For every non-zero x, y, z∈X with z /∈V(x, y), we have (kx, zk+ky, zk)

x

kx, zk − y ky, zk, z

≤2kx−y, zk. (2.12)

Proof. Since

kx, zk

ky, zk +ky, zk kx, zk ≥2, we have the inequality

(kx, zk+ky, zk)²−(x, y|z)

kx, zk

ky, zk +ky, zk kx, zk

−2 (x, y|z)

≤2kx, zk²+ky, zk²−4 (x, y|z) which implies (2.12).

Theorem 2.8. For every non-zero x, y, z∈X with z /∈V(x, y)we have

(kx, zk+ky, zk)² x

kx, zk − y ky, zk, z

2

+

x

kx, zk + y ky, zk, z

2! (2.13)

≤8

kx, zk²+ky, zk² .

Proof. By (2.12) we have (kx, zk+ky, zk)²

x

kx, zk− y ky, zk, z

2

+

x

kx, zk + y ky, zk, z

2!

≤ 4

kx−y, zk²+kx+y, zk²

and, by a 2−dimensional analogue of the parallelogram law, we get (2.13). Remark 2.2. For some similar results in inner product spaces, see [7].

3. A´czel’s Inequality

In this section, we shall point out some results in 2−inner product spaces in con- nection to A´czel’s inequality [12]. For some other similar results in inner products, see [8]. We note that the results obtained here, in 2−inner product spaces used different techniques as those in [8].

Theorem 3.1. Let (X,(·,· | ·)) be a 2−inner product space, M1, M2 ∈ R and x, y, z∈X such that

kx, zk ≤ |M1|, ky, zk ≤ |M2|,

then

M₁²− kx, zk² M₂²− ky, zk²

≤(|M₁M₂| −(x, y|z))². (3.1)

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Proof. Using the elementary inequality (2.3),we get 0≤

M₁²− kx, zk² M₂²− ky, zk²

≤(|M1M2| − kx, zk ky, zk)², and by Cauchy-Schwarz’s inequality,

0≤ |M1M2| − kx, zk ky, zk ≤ |M1M2| −(x, y|z) implying (3.1).

Corollary 3.2. If x, y, z ∈X, are such that kx, zk,ky, zk ≤ M, M > 0, then we have the inequality

0≤ kx, zk²ky, zk²−(x, y|z)²≤M²kx−y, zk² (3.2)

which is a counterpart of Cauchy-Schwarz’s inequality.

Another similar results to the generalization (3.1) of A´czel’s inequality is the following one

(|M₁| − kx, zk)¹²(|M₂| − ky, zk)¹² ≤ |M₁M₂|¹² − |(x, y|z)|¹². (3.3)

Proof. Applying (2.3) form=p

|M1|, p=p

|M2|, n=p

kx, zk, q=p ky, zk and using Cauchy-Schwarz’s inequality for 2−inner products we deduce (3.3). Corollary 3.4. Suppose thatx, y, z∈X andM >0are such thatkx, zk,ky, zk ≤ M. Then we have the following converse of Cauchy-Schwarz’s inequality

0≤ kx, zk ky, zk − |(x, y|z)| (3.4)

≤M

kx, zk+ky, zk −2|(x, y|z)|^1/2 .

Theorem 3.5. Let (·,· | ·)be a 2−inner product and {(·,· | ·)_i}i∈N a sequence of 2−inner products satisfying

kx, zk²>

X∞ i=0

kx, zk²i

(3.5)

for all x, z,being linearly independent. Then we have the following refinemenet of Cauchy-Schwarz’s inequality

kx, zk ky, zk − |(x, y|z)| (3.6)

≥

"_∞ X

i=0

kx, zki

X∞ i=0

ky, zki− |(x, y|z)|

#

≥0 (3.7)

for allx, y, z∈X.

Proof. Letn∈Nandn≥1.Define the mapping (x, y|z)_n = (x, y|z)−

Xn

i=0

(x, y|z)_i, x, y, z∈X.

We observe, by (3.5),that the mapping (·,· | ·)_n satisfies the properties (i) (x, x|z)_n≥0,

(ii) (αx+βx⁰, y|z)_n=α(x, y|z)_n+β(x⁰, y|z)_n,

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(iii) (x, y |z)_n = (y, x|z)_n

for everyx, x⁰, y, z∈X andα, α⁰∈R.

By a similar proof to that in Theorem 2.4, we can state Cauchy-Schwarz’s inequality

(x, x|z)_n(y, y|z)_n≥ |(x, y|z)_n|², x, y, z∈X, that is

kx, zk²− Xn

i=0

kx, zk²i

!

ky, zk²− Xn

i=0

ky, zk²i

! (3.8)

≥ (x, y|z)− Xn

i=0

(x, y|z)_i

!2

.

Using A´czel’s inequality [12]

a²− Xm

i=0

a²_i

! b²−

Xm

i=0

b²_i

!

≤ ab− Xm

i=0

aibi

!2

,

wherea, b, a_i, b_i∈Rfori= 0, ..., m; we can prove that kx, zk ky, zk −

Xn

i=0

kx, zkiky, zki

!2

(3.9)

≥ kx, zk²− Xn

i=0

kx, zk²i

!

ky, zk²− Xn

i=0

ky, zk²i

!

for allx, y, z∈X.Since, by Cauchy-Buniakowski-Schwarz’s inequality kx, zk ky, zk ≥

Xn

i=0

kx, zk²i

Xn

i=0

ky, zk²i

!1/2

≥ Xn

i=0

kx, zkiky, zki,

then by (3.8) and (3.9) we deduce kx, zk ky, zk −

Xn

i=0

kx, zkiky, zki

=

kx, zk ky, zk − Xn

i=0

kx, zkiky, zki

≥ |(x, y|z)| − Xn

i=0

|(x, y|z)_i| which implies (3.6), by using the inequality

kx, zkiky, zki− |(x, y|z)_i| ≥0.

The theorem is thus proved.

The following corollaries are interesting as refinements of the triangle inequality for 2-norms generated by 2-inner products.

Corollary 3.6. With the assumptions from Theorem, we have the following re- finement of the triangle inequality

(kx, zk+ky, zk)²− kx+y, zk²

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≥ X∞

i=0

h

(kx, zki+ky, zki)²− kx+y, zk²i

i

≥0, x, y, z∈X.

Corollary 3.7. Let(., .|.)₁,(., .|.)₂ be two 2-inner products such that kx, zk2>kx, zk1

for allx, z being linearly independent in X.Then

kx, zk2ky, zk2− |(x, y|z)₂|

≥ kx, zk1ky, zk1− |(x, y|z)₁| ≥0, x, y, z ∈X.

Corollary 3.8. Let(., .|.)₁,(., .|.)₂ be as above. Then (kx, zk2+ky, zk2)²− kx+y, zk²2

≥(kx, zk1+ky, zk1)²− kx+y, zk²1≥0, x, y, z∈X.

References

[1] Y.J. Cho, S.S. Kim, Gˆateaux derivatives and 2-inner product spaces,Glaˇsnik Mat.,27(1992), 271-282.

[2] Y.J. Cho, C.S. Lin, A. Misiak,Theory of 2-Inner Product Spaces, unpublished book [3] C. Diminnie, S. G¨ahler, A. White, 2-inner product spaces, Demonstratio Mathematica,

6(1973), 525-536.

[4] , Remarks on generalizations of 2-inner products,Math. Nachr.,74(1974), 269-278.

[5] , 2-inner product spaces II,Demonstratio Mathematica,10(1977), 169-188.

[6] C. Diminnie, A. White, 2-inner product spaces and Gˆateaux partial derivatives,Comment.

Math. Univ. Carolinae,16(1975), 115-119.

[7] S.S. Dragomir, I. S´andor, Some inequalities in prehilbertian spaces, Studia Univ. Babes- Bolyai, Mathematica,32(1987), 71-78.

[8] S.S. Dragomir, A generalization of A´czel’s inequality in inner product spaces, Acta Math.

Hungar.,65(1994), 141-148.

[9] R. Ehret, Linear 2-normed spaces,Doctoral Diss., Saint Louis Univ., 1969.

[10] S. G¨ahler, Lineare 2-normierte R¨aume,Math. Nachr.,28(1965), 1-43.

[11] S. G¨ahler, A. Misiak, Remarks on 2-inner products,Demonstratio Math.,17(1984), 655-670.

[12] D.S. Mitrinovi´c,Analytic Inequalities, Springer Verlag, 1970.

Department of Mathematics, Gyeonsang National University, Chinju 660-701, Korea E-mail address, Y.J. Cho: [email protected]

School of Communications and Informatics, Victoria University of Technology, PO Box 14428, Melbourne City MC, Victoria 8001, Australia.

E-mail address, S.S. Dragomir: [email protected]

Department of Mathematics, St. Bonaventure University, St. Bonaventure, NY 14778, U.S.A.

Department of Mathematics, Dongeui University, Pusan 614-714