On Inequalities in Normed Linear Spaces and Applications

On Inequalities in Normed Linear Spaces and Applications

This is the Published version of the following publication

Dragomir, Sever S, Koliha, Jerry J and Cho, Yeol Je (2000) On Inequalities in Normed Linear Spaces and Applications. RGMIA research report collection, 3 (1).

The publisher’s official version can be found at

Note that access to this version may require subscription.

Downloaded from VU Research Repository https://vuir.vu.edu.au/17290/

LINEAR SPACES AND APPLICATIONS

S. S. Dragomir, J. J. Koliha and Y. J. Cho

Abstract. In this paper we obtain inequalities involving the norms of finite se- quences of vectors in normed linear spaces, which are new even for complex num- bers.

I. Introduction

Let (X,k · k) be a normed linear space. The following inequality is well known in the literature as the polygonal inequality, which is a generalization of the triangle inequality:

(P) X

i∈I

xi



≤X

i∈I

kxik;

I is a finite set of indices and x_i (i ∈I) are vectors in X.

In papers [1–3], authors explored relations between the following refinements and generalizations of (P):

Theorem A. Let (X,k · k) be a normed linear space and I be a finite set of indices,xi∈X, i∈I and p∈R, p≥1. Then we have the following inequalities:

(1)



X

i∈I

xi



^p ≤[card(I)]^p−k−1 X

i1,...,ik+1∈I





xi1 +· · ·+xik+1

k+ 1





p

≤[card(I)]^p−k X

i1,...,ik∈I





x_i1 +· · ·+x_ik k





p

≤ · · ·

≤[card(I)]^p−2 X

i1,i2∈I



x_i1 +x_i2 2





p

≤[card(I)]^p−1X

i∈I

kx_ik^p,

Typeset byAMS-TEX 1

2 S. S. DRAGOMIR, J. J. KOLIHA AND Y. J. CHO

where k ∈N, k ≥1.

The following result is a refinement of (P) for weighted mean:

Theorem B. With the above assumptions, if g_j (j = 1,2,3, . . . , k) are nonneg- ative weights such that Pk

j=1gj >0, then we have the following inequality:

(2)



X

i∈I

x_i^p ≤[card(I)]^p−k X

i1,...,ik∈I



x_i1+· · ·+x_ik k





p

≤[card(I)]^p−k X

i1,...,ik∈I



g1xi1 +· · ·+gkxik

g₁+· · ·+g_k





p

≤[card(I)]^p−1X

i∈I

kx_ik^p.

Note that both the above inequalities were obtained from more general results for convex mappings defined on convex subsets in normed linear spaces.

In this paper, using only techniques from normed linear spaces, we obtain new inequalities for the norms of finite sequences.

II. The Results

We start with the following theorem:

Theorem 2.1. Let (X,k · k) be a normed linear space, x_i ∈ X with i ∈ I (I is finite) and x=P

i∈Ixi. Then we have the following inequalities:

(3) k xkx−X

i∈I

kx±x_ikx_i≤X

i∈I

kx_ik² ≤ ¹₂

kxk+X

i∈I

kx_ik‘ X

i∈I

kx_ik.

Proof. For a fixed i∈I, we have kx_ik=x− X

j∈I,j6=i

x_j≤ kxk+ X

j∈I,j6=i

x_j

≤ kxk+ X

j∈I,j6=i

kx_jk=kxk+X

j∈I

kx_jk − kx_ik,

which implies that, for all i ∈I, kx_ik ≤ ¹₂

kxk+X

j∈I

kx_jk‘ .

Multiplying by kx_ik ≥0 and summing over i∈I, we deduce X

i∈I

kx_ik² ≤ ¹₂

kxk+X

j∈I

kx_jk‘ X

i∈I

kx_ik,

and so the second inequality in (1) is proved. Also, for a fixed i∈I, we have kxik=k −xik=kx±xi−xk ≥ŒŒŒkx±xik − kxkŒŒŒ.

Multiplying by kx_ik ≥0, we have

kx_ik² ≥ŒŒŒkx±x_ik − kxkŒŒŒkx_ik=(kx±x_ik − kxk)x_i for all i∈I. Summing over i∈I, we have

X

i∈I

kxik² ≥X

i∈I



kx±xikxi− kxkxi



≥X

i∈I

kx±xikxi− kxkxi



,

and so the theorem is proved.

Corollary 2.2. With the above assumptions, if P

i∈Ix_i = 0, then

(4) X

i∈I

kx_ik² ≤ ¹₂ X

i∈I

kx_ik‘2

.

The following result generalizes Theorem 2.1. In its proof we need the formula

(5) X

i1,...,ik∈I

(x_i1 +· · ·+x_ik) =k[card(I)]^k−1X

i∈I

x_i,

which can be proved by induction on k.

Theorem 2.3. With the above assumptions, we have the following inequalities:

(6)

k X

i1,...,ik∈I

kx±(x_i1 +· · ·+x_ik)kx_i1 −n^k−1kxkx

≤ X

i1,...,ik∈I

kx_i1+· · ·+x_ikk²

≤ ¹₂ X

i1,...,ik∈I

kx_i1 +· · ·+x_ikk²+k X

i1,...,ik∈I

kx_i1k kx_i1 +· · ·+x_ikk‘

≤ ¹₂

kxk+X

i∈I

kx_ik‘ X

i1,...,ik∈I

kx_i1+· · ·+x_ikk,

4 S. S. DRAGOMIR, J. J. KOLIHA AND Y. J. CHO

where k ≥1 and card(I) =n.

Proof. For fixed i1, . . . , ik ∈I, we have kx_i1 +· · ·+x_ikk=x− X

j /∈{i1,...,ik}

x_j

≤ kxk+ X

j /∈{i1,...,ik}

x_j

≤ kxk+ X

j /∈{i1,...,ik}

kx_jk

=kxk+X

i∈I

kxik − kxi1k − · · · − kxikk,

from which it follows

kxi1 +· · ·+xikk+kxi1k+· · ·+kxikk ≤ kxk+X

i∈I

kxik.

Multiplying by kxi1 +· · ·+xikk ≥0 and summing over i1, . . . , ik in I, we have X

i1,...,ik∈I

kx_i1+· · ·+x_ikk²+k X

i1,...,ik∈I

kx_i1k kx_i1 +· · ·+x_ikk

≤

kxk+X

i∈I

kx_ik‘ X

i1,...,ik∈I

kx_i1 +· · ·+x_ikk,

and so the third inequality in (6) follows. By the triangle inequality kx_i1 +· · ·+x_ikk ≤ ¹₂(kx_i1 +· · ·+x_ikk+kx_i1k+· · ·+kx_ikk).

Multiplying by kx_i1 +· · ·+x_ikk and summing over i₁, . . . , i_k in the index set I, we derive

X

i1,...,ik∈I

kx_i1 +· · ·+x_ikk²

≤ ¹₂ X

i1,...,ik∈I

kxi1 +· · ·+xikk²+k X

i1,...,ik∈I

kxi1kkx_i1 +· · ·+x_ikk‘

and so the second inequality in (6) is also proved.

Finally, we prove the first inequality in (6). Since kx_i1 +· · ·+x_ikk=k −(x_i1 +· · ·+x_ik)k

=kx±(x_i1 +· · ·+x_ik)−xk

≥ŒŒŒkx±(x_i1 +· · ·+x_ik)k − kxkŒŒŒ,

then

kx_i1 +· · ·+x_ikk² ≥(kx±(x_i1 +· · ·+x_ik)k − kxk)(x_i1 +· · ·+x_ik)

=kx±(x_i1 +· · ·+x_ik)k(x_i1 +· · ·+x_ik)− kxk(x_i1 +· · ·+x_ik). Summing over i₁, . . . , i_k in the index set I, we use (5) to obtain

X

i1,...,ik∈I

kx_i1 +· · ·+x_ikk²

≥ X

i1,...,ik∈I



kx±(xi1 +· · ·+xik)k(x_i1 +· · ·+x_ik)− kxk(x_i1 +· · ·+x_ik)

≥k X

i1,...,ik∈I

kx±(x_i1 +· · ·+x_ik)kx_i1 −kn^k−1kxkX

i∈I

x_i

=k X

i1,...,ik∈I

kx±(xi1 +· · ·+xik)kxi1 −n^k−1)kxkx

and thus the theorem is proved.

Corollary 2.4. With the above assumptions, if P

i∈Ix_i = 0, then

(7)

k X

i1,...,ik∈I

kx_i1 +· · ·+x_ikkx_i1

≤ X

i1,...,ik∈I

kx_i1 +· · ·+x_ikk²

≤ ¹₂ X

i1,...,ik∈I

kx_i1 +· · ·+x_ikk²+k X

i1,...,ik∈I

kx_i1k kx_i1 +· · ·+x_ikk‘

≤ ¹₂X

i∈I

kx_i1k X

i1,...,ik∈I

kx_i1 +· · ·+x_ikk.

III. Applications to Complex Numbers

6 S. S. DRAGOMIR, J. J. KOLIHA AND Y. J. CHO

The inequalities obtained in Section II can be restated to obtain new inequal- ities for complex numbers:

1. Let z_i ∈C, i∈I (I is finite) and let z =P

i∈Iz_i. Then

ŒŒ

Œ|z|z −X

i∈I

|z ±z_i|z_iŒŒŒ≤X

i∈I

|z_i|² ≤ ¹₂

|z|+X

i∈I

|z_i|‘ X

i∈I

|z_i|.

In particular, if z = 0, then X

i∈I

|z_i|² ≤ ¹₂ X

i∈I

|z_i|‘2

.

2. With the above assumptions, we have kŒŒŒ X

i1,...,ik∈I

|z±(z_i1 +· · ·+z_ik)|z_i1 −n^k−1|z|zŒŒŒ

≤ X

i1,...,ik∈I

(zi1 +· · ·+zik)²

≤ ¹₂ X

i1,...,ik∈I

|z_i1 +· · ·+x_ik|²+k X

i1,...,ik∈I

|z_i1| |z_i1 +· · ·+z_ik|‘

≤ ₂¹

|z|+X

i∈I

|z_i|‘ X

i1,...,ik∈I

|z_i1 +· · ·+z_ik|,

where k ∈N, k ≥1 and card(I) =n.

In particular, ifz = 0, we have kŒŒŒ X

i1,...,ik∈I

|z_i1 +· · ·+z_ik|z_i1ŒŒŒ≤ X

i1,...,ik∈I

|z_i1 +· · ·+z_ik|²

≤ ¹₂ X

i1,...,ik∈I

|z_i1 +· · ·+x_ik|² +k X

i1,...,ik∈I

|z_i1(z_i1 +· · ·+z_ik)|‘

≤ ¹₂X

i∈I

|zi| X

i1,...,ik∈I

|zi1 +· · ·+zik|.

References

1. S. S. Dragomir,Some refinements of Ky Fan’s inequality, J. Math. Anal. Appl.163(1992), 317–321.

2. S. S. Dragomir,On some refinements of Jensen’s inequality and applications, Utilitas Math.

43(1993), 235–243.

3. J. E. Pecaric and S. S. Dragomir, A refinement of Jensen’s inequality and applications, Studia Univ. Babes-Bolyai Math. 34(1989), 15–19.

School of Communications and Informatics, Victoria University of Technol- ogy, Melbourne, VIC 8001, Australia

Department of Mathematics and Statistics, University of Melbourne, VIC 3010 Australia

Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea