REVERSE TRIANGLE INEQUALITY IN HILBERT C∗-MODULES

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Reverse Triangle Inequality Maryam Khosravi, Hakimeh Mahyar

and Mohammad Sal Moslehian vol. 10, iss. 4, art. 110, 2009

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REVERSE TRIANGLE INEQUALITY IN HILBERT C

^∗

-MODULES

MARYAM KHOSRAVI, HAKIMEH MAHYAR

Department of Mathematics Tarbiat Moallem University Tahran, Iran.

EMail:{khosravi_m,mahyar}@saba.tmu.ac.ir

MOHAMMAD SAL MOSLEHIAN

Department of Pure Mathematics

Centre of Excellence in Analysis on Algebraic Structures (CEAAS) Ferdowsi University of Mashhad

P.O. Box 1159, Mashhad 91775, Iran.

EMail:moslehian@ferdowsi.um.ac.ir

Received: 16 October, 2009

Accepted: 13 November, 2009

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: Primary 46L08; Secondary 15A39, 26D15, 46L05, 51M16.

Key words: Triangle inequality, Reverse inequality, HilbertC^∗-module,C^∗-algebra.

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Close Abstract: We prove several versions of reverse triangle inequality in Hilbert C^∗-

modules. We show that ife₁, . . . , e_mare vectors in a Hilbert moduleX over aC^∗-algebraAwith unit 1 such thathe_i, e_ji= 0 (1≤i6=j ≤m) andke_ik = 1 (1 ≤ i ≤ m), and alsor_k, ρ_k ∈ R(1 ≤ k ≤ m)and x1, . . . , xn∈Xsatisfy

0≤r_k²kxjk ≤Rehrkek, xji, 0≤ρ²_kkxjk ≤Imhρkek, xji,

then

"_m X

k=1

r²_k+ρ²_k

#¹₂ _n X

j=1

kxjk ≤

n

X

j=1

xj

,

and the equality holds if and only if

n

X

j=1

xj=

n

X

j=1

kxjk

m

X

k=1

(rk+iρk)ek.

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1. Introduction and Preliminaries

The triangle inequality is one of the most fundamental inequalities in mathematics.

Several mathematicians have investigated its generalizations and its reverses.

In 1917, Petrovitch [17] proved that for complex numbersz₁, . . . , z_n, (1.1)

n

X

j=1

z_j

≥cosθ

n

X

j=1

|z_j|,

where0< θ < ^π₂ andα−θ <arg z_j < α+θ (1≤j ≤n)for a given real number α.

The first generalization of the reverse triangle inequality in Hilbert spaces was given by Diaz and Metcalf [5]. They proved that for x1, . . . , xn in a Hilbert space H, if eis a unit vector of H such that 0 ≤ r ≤ ^Rehx_kx^j^,ei

jk for somer ∈ Rand each 1≤j ≤n, then

(1.2) r

n

X

j=1

kx_jk ≤

n

X

j=1

x_j . Moreover, the equality holds if and only ifPn

j=1x_j =rPn

j=1kx_jke.

Recently, a number of mathematicians have presented several refinements of the reverse triangle inequality in Hilbert spaces and normed spaces (see [1, 2, 4, 7, 8, 10,13,16]). Recently a discussion ofC^∗-valued triangle inequalities in HilbertC^∗- modules was given in [3]. Our aim is to generalize some of the results of Dragomir in Hilbert spaces to the framework of HilbertC^∗-modules. For this purpose, we first recall some fundamental definitions in the theory of HilbertC^∗-modules.

Suppose that A is a C^∗-algebra and X is a linear space, which is an algebraic rightA-module. The space Xis called a pre-HilbertA-module (or an inner product A-module) if there exists anA-valued inner product h·,·i : X×X → A with the following properties:

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(i) hx, xi ≥0andhx, xi= 0if and only ifx= 0;

(ii) hx, λy+zi=λhx, yi+hx, zi;

(iii) hx, yai=hx, yia;

(iv) hx, yi^∗ =hy, xi

for all x, y, z ∈ X, a ∈ A, λ ∈ C. By (ii) and (iv), h·,·i is conjugate linear in the first variable. Using the Cauchy–Schwartz inequalityhy, xihx, yi ≤ khx, xikhy, yi [11, Page 5] (see also [14]), it follows that kxk = khx, xik¹² is a norm on Xmaking it a right normed module. The pre-Hilbert module X is called a Hilbert A-module if it is complete with respect to this norm. Notice that the inner structure of a C^∗-algebra is essentially more complicated than that for complex numbers. For instance, properties such as orthogonality and theorems such as Riesz’ representation in complex Hilbert space theory cannot simply be generalized or transferred to the theory of HilbertC^∗-modules.

One may define an “A-valued norm”|·|by|x|=hx, xi^1/2. Clearly,k |x| k=kxk for eachx∈X. It is known that|·|does not satisfy the triangle inequality in general.

See [11,12] for more information on HilbertC^∗-modules.

We also use elementary C^∗-algebra theory, in particular we utilize the property that ifa ≤ b thena^1/2 ≤ b^1/2, wherea, bare positive elements of aC^∗-algebraA.

We also repeatedly apply the following known relation:

(1.3) 1

2(aa^∗+a^∗a) = (Rea)² + (Ima)²,

where a is an arbitrary element of A. For details on C^∗-algebra theory, we refer readers to [15].

Throughout the paper, we assume thatA is a unital C^∗-algebra with unit1 and for everyλ∈C, we writeλforλ1.

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2. Multiplicative Reverse of the Triangle Inequality

Utilizing someC^∗-algebraic techniques we present our first result as a generalization of [7, Theorem 2.3].

Theorem 2.1. LetAbe aC^∗-algebra, letXbe a HilbertA-module and letx₁, . . . , x_n∈ X. If there exist real numbersk1, k2 ≥0with

0≤k₁kx_jk ≤Rehe, x_ji, 0≤k₂kx_jk ≤Imhe, x_ji, for somee∈Xwith|e| ≤1and all1≤j ≤n, then

(2.1) (k²₁+k₂²)¹²

n

X

j=1

kx_jk ≤

n

X

j=1

x_j .

Proof. Applying the Cauchy–Schwarz inequality, we get

* e,

n

X

j=1

x_j +

2

≤ kek²

n

X

j=1

x_j

2

≤

n

X

j=1

x_j

2

, and

* _n X

j=1

x_j, e +

2

≤

n

X

j=1

x_j

2

|e|² ≤

n

X

j=1

x_j

2

,

whence

n

X

j=1

x_j

2

≥ 1 2





* e,

n

X

j=1

x_j +

2

+

* _n X

j=1

x_j, e +

2



= 1 2

* e,

n

X

j=1

x_j +∗*

e,

n

X

j=1

x_j +

+

* _n X

j=1

x_j, e

+∗* _n X

j=1

x_j, e +!

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= Re

* e,

n

X

j=1

x_j +!2

+ Im

* e,

n

X

j=1

x_j +!2

by(1.3)

= Re

n

X

j=1

he, x_ji

!2

+ Im

n

X

j=1

he, x_ji

!2

≥k₁²

n

X

j=1

kx_jk

!2

+k²₂

n

X

j=1

kx_jk

!2

= (k²₁+k₂²)

n

X

j=1

kx_jk

!2

.

Using the same argument as in the proof of Theorem 2.1, one can obtain the following result, wherek₁, k₂are hermitian elements ofA.

Theorem 2.2. If the vectorsx1, . . . , xn ∈Xsatisfy the conditions

0≤k²₁kxjk² ≤(Rehe, xji)², 0≤k₂²kxjk² ≤(Imhe, xji)²,

for some hermitian elementsk₁, k₂inA, somee∈Xwith|e| ≤1and all1≤j ≤n then the inequality (2.1) holds.

One may observe an integral version of inequality (2.1) as follows:

Corollary 2.3. Suppose thatXis a HilbertA-module andf : [a, b]→Xis strongly measurable such that the Lebesgue integralRb

a kf(t)kdt exists and is finite. If there exist self-adjoint elementsa₁, a₂inAwith

a²₁kf(t)k² ≤Rehf(t), ei², a²₂kf(t)k² ≤Imhf(t), ei² (a.e. t∈[a, b]),

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wheree ∈Xwith|e| ≤1, then (a²₁ +a²₂)¹²

Z b a

kf(t)kdt≤

Z b a

f(t)dt .

Now we prove a useful lemma, which is frequently applied in the next theorems (see also [3]).

Lemma 2.4. LetXbe a HilbertA-module and let x, y ∈ X. If |hx, yi| = kxkkyk, then

y= xhx, yi kxk² . Proof. Forx, y ∈Xwe have

0≤

y− xhx, yi kxk²

2

=

y− xhx, yi

kxk² , y−xhx, yi kxk²

=hy, yi − 1

kxk²hy, xihx, yi+ 1

kxk⁴hy, xihx, xihx, yi − 1

kxk²hy, xihx, yi

≤ |y|²− 1

kxk²|hx, yi|² =|y|²− 1

kxk²kxk²kyk²

=|y|²− kyk² ≤0, whence

y− ^xhx,yi_kxk2

= 0. Hencey= ^xhx,yi_kxk2 .

Using the Cauchy–Schwarz inequality, we have the following theorem for Hilbert modules, which is similar to [1, Theorem 2.5].

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Theorem 2.5. Let e₁, . . . , e_m be a family of vectors in a Hilbert module Xover a C^∗-algebraAsuch thathe_i, e_ji = 0 (1≤ i 6=j ≤ m)andke_ik = 1 (1 ≤i ≤ m).

Suppose thatr_k, ρ_k ∈R(1≤k ≤m)and that the vectorsx₁, . . . , x_n∈Xsatisfy 0≤r_k²kx_jk ≤Rehr_ke_k, x_ji, 0≤ρ²_kkx_jk ≤Imhρ_ke_k, x_ji,

Then

(2.2)

" _m X

k=1

(r²_k+ρ²_k)

#¹₂ _n X

j=1

kx_jk ≤

n

X

j=1

x_j ,

and the equality holds if and only if (2.3)

n

X

j=1

x_j =

n

X

j=1

kx_jk

m

X

k=1

(r_k+iρ_k)e_k.

Proof. There is nothing to prove ifPm

k=1(r²_k+ρ²_k) = 0. Assume thatPm

k=1(r²_k+ ρ²_k)6= 0. From the hypothesis, byIm(a) = Re(ia^∗),Re(a^∗) = Re(a) (a ∈A),we have

m

X

k=1

(r²_k+ρ²_k)

!2 n

X

j=1

kx_jk

!2

≤ Re

* _m X

k=1

rkek,

n

X

j=1

xj

+ + Im

* _m X

k=1

ρkek,

n

X

j=1

xj

+!2

= Re

* _n X

j=1

x_j,

m

X

k=1

(r_k+iρ_k)e_k +!2

.

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By (1.3),

Re

* _n X

j=1

x_j,

m

X

k=1

(r_k+iρ_k)e_k +!2

≤ 1 2





* _n X

j=1

x_j,

m

X

k=1

(r_k+iρ_k)e_k +

2

+

* _m X

k=1

(r_k+iρ_k)e_k,

n

X

j=1

x_j +

2



≤ 1 2





n

X

j=1

x_j

2

m

X

k=1

(r_k+iρ_k)e_k

2

+

m

X

k=1

(r_k+iρ_k)e_k

2

n

X

j=1

x_j

2



≤

n

X

j=1

x_j

2

m

X

k=1

(r_k+iρ_k)e_k

2

and since|a| ≤ kak (a∈A),

n

X

j=1

x_j

2

m

X

k=1

(r_k+iρ_k)e_k

2

≤

n

X

j=1

x_j

2

* _m X

k=1

(r_k+iρ_k)e_k,

m

X

k=1

(r_k+iρ_k)e_k +

=

n

X

j=1

xj

2 m

X

k=1

|rk+iρk|²kekk²

=

n

X

j=1

x_j

2 m

X

k=1

(r_k²+ρ²_k).

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Hence

" _m X

k=1

(r²_k+ρ²_k)

# _n X

j=1

kx_jk

!2

≤

n

X

j=1

x_j

2

. By taking square roots the desired result follows.

Clearly we have equality in (2.2) if condition (2.3) holds. To see the converse, first note that if equality holds in (2.2), then all inequalities in the relations above should be equality. Therefore

r_k²kxjk= Rehrkek, xji, ρ²_kkxjk= Imhρkek, xji, Re

* _n X

j=1

x_j,

m

X

k=1

(r_k+iρ_k)e_k +

=

* _n X

j=1

x_j,

m

X

k=1

(r_k+iρ_k)e_k +

,

and

* _m X

k=1

(r_k+iρ_k)e_k,

n

X

j=1

x_j +

=

n

X

j=1

x_j

m

X

k=1

(r_k+iρ_k)e_k .

From Lemma2.4and the above equalities we have

n

X

j=1

x_j = Pm

k=1(r_k+iρ_k)e_k kPm

k=1(r_k+iρ_k)e_kk²

* _m X

k=1

(r_k+iρ_k)e_k,

n

X

j=1

x_j +

= Pm

k=1(r_k+iρ_k)e_k Pm

k=1(r²_k+ρ²_k) Re

* _m X

k=1

(r_k+iρ_k)e_k,

n

X

j=1

x_j +

= Pm

k=1(r_k+iρ_k)e_k Pm

k=1(r²_k+ρ²_k)

m

X

k=1 n

X

j=1

(r²_kkxjk+ρ²_kkxjk)

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=

n

X

j=1

kx_jk

m

X

k=1

(r_k+iρ_k)e_k,

which is the desired result.

In the next results of this section, we assume thatXis a right HilbertA-module, which is an algebraic leftA-module subject to

hx, ayi=ahx, yi (x, y ∈X, a∈A). (†) For example ifAis a unitalC^∗-algebra andIis a commutative right ideal ofA, then Iis a right Hilbert module overAand

hx, ayi=x^∗(ay) = ax^∗y=ahx, yi (x, y ∈I, a∈A).

The next theorem is a refinement of [7, Theorem 2.1]. To prove it we need the following lemma.

Lemma 2.6. LetXbe a HilbertA-module ande₁, . . . , e_n ∈Xbe a family of vectors such thathe_i, e_ji= 0 (i6=j)andke_ik= 1. Ifx∈X, then

|x|² ≥

n

X

k=1

|he_k, xi|² and |x|² ≥

n

X

k=1

|hx, e_ki|². Proof. The first result follows from the following inequality:

0≤

x−

n

X

k=1

e_khe_k, xi

2

=

* x−

n

X

k=1

e_khe_k, xi, x−

n

X

j=1

e_jhe_j, xi +

=hx, xi+

n

X

k=1 n

X

j=1

he_k, xi^∗he_k, e_jihe_j, xi −2

n

X

k=1

|he_k, xi|²

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=hx, xi+

n

X

k=1

he_k, xi^∗he_k, e_kihe_k, xi −2

n

X

k=1

|he_k, xi|²

≤ |x|²+

n

X

k=1

he_k, xi^∗he_k, xi −2

n

X

k=1

|he_k, xi|²

=|x|² −

n

X

k=1

|he_k, xi|². By considering|x−Pn

k=1hek, xiek|², we similarly obtain the second one.

Now we will prove the next theorem without using the Cauchy–Schwarz inequality.

Theorem 2.7. Let e₁, . . . , e_m ∈ X be a family of vectors with he_i, e_ji = 0 (1 ≤ i 6= j ≤ m)andke_ik = 1 (1 ≤ i ≤ m). If the vectors x₁, . . . , x_n ∈ X satisfy the conditions

(2.4) 0≤r_kkx_jk ≤Rehe_k, x_ji, 0≤ρ_kkx_jk ≤Imhe_k, x_ji, for1≤j ≤n,1≤k ≤m, wherer_k, ρ_k ∈[0,∞) (1 ≤k≤m), then

(2.5)

" _m X

k=1

(r²_k+ρ²_k)

#¹₂ _n X

j=1

kx_jk ≤

n

X

j=1

x_j .

Proof. Applying the previous lemma forx=Pn

j=1xj, we obtain

n

X

j=1

x_j

2

≥ 1 2





m

X

k=1

* e_k,

n

X

j=1

x_j +

2

+

m

X

k=1

* _n X

j=1

x_j, e_k +

2



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=

m

X

k=1

1 2

* e_k,

n

X

j=1

x_j +^∗*

e_k,

n

X

j=1

x_j +

+

* _n X

j=1

x_j, e_k

+^∗* _n X

j=1

x_j, e_k +!

=

m

X

k=1

Re

* e_k,

n

X

j=1

x_j +!2

+ Im

* e_k,

n

X

j=1

x_j +!2

(by (1.3))

=

m

X

k=1

Re

n

X

j=1

he_k, x_ji

!2

+ Im

n

X

j=1

he_k, x_ji

!2

≥

m

X

k=1



r²_k

n

X

j=1

kx_jk

!2

+ρ²_k

n

X

j=1

kx_jk

!2

 (by (2.4))

=

m

X

k=1

(r²_k+ρ²_k)

n

X

j=1

kx_jk

!2

.

Proposition 2.8. In Theorem2.7, ifhe_k, e_ki = 1, then the equality holds in (2.5) if and only if

(2.6)

n

X

j=1

xj =

n

X

j=1

kxjk

! _m X

k=1

(rk+iρk)ek.

Proof. If (2.6) holds, then the inequality in (2.5) turns trivially into equality.

Next, assume that equality holds in (2.5). Then the two inequalities in the proof

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of Theorem2.7should be equalities. Hence

n

X

j=1

x_j

2

=

m

X

k=1

* e_k,

n

X

j=1

x_j +

2

and

n

X

j=1

x_j

2

=

m

X

k=1

* _n X

j=1

x_j, e_k +

2

,

which is equivalent to

n

X

j=1

x_j =

m

X

k=1 n

X

j=1

e_khe_k, x_ji=

m

X

k=1 n

X

j=1

he_k, x_jie_k,

and

rkkxjk= Rehek, xji, ρkkxjk= Imhek, xji. So

n

X

j=1

x_j =

m

X

k=1 n

X

j=1

e_khe_k, x_ji

=

m

X

k=1 n

X

j=1

e_k(r_k+iρ_k)kx_jk

=

n

X

j=1

kx_jk

! _m X

k=1

(r_k+iρ_k)e_k.

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3. Additive Reverse of the Triangle Inequality

We now present some versions of the additive reverse of the triangle inequality. In [6], Dragomir established the following theorem:

Theorem 3.1. Let{e_k}^m_k=1 be a family of orthonormal vectors in a Hilbert spaceH andM_jk ≥0 (1≤j ≤n,1≤k ≤m)such that

kxjk −Rehek, xji ≤Mjk, for each1≤j ≤nand1≤k ≤m. Then

n

X

j=1

kx_jk ≤ 1

√m

n

X

j=1

x_j

+ 1 m

n

X

j=1 m

X

k=1

M_jk;

and the equality holds if and only if

n

X

j=1

kxjk ≥ 1 m

n

X

j=1 m

X

k=1

Mjk,

and n

X

j=1

x_j =

n

X

j=1

kx_jk − 1 m

n

X

j=1 m

X

k=1

M_jk

! _m X

k=1

e_k.

We can prove this theorem for Hilbert C^∗-modules using some different techniques.

Theorem 3.2. Let{e_k}^m_k=1 be a family of vectors in a Hilbert moduleXover a C^∗- algebraA with unit1, |e_k| ≤ 1 (1 ≤ k ≤ m), he_i, e_ji = 0 (1 ≤ i 6= j ≤ m)and x_j ∈X (1≤j ≤n). If for some scalarsM_jk ≥0 (1≤j ≤n,1≤k ≤m),

(3.1) kxjk −Rehek, xji ≤Mjk (1≤j ≤n,1≤k≤m),

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then (3.2)

n

X

j=1

kx_jk ≤ 1

√m

n

X

j=1

x_j

+ 1 m

n

X

j=1 m

X

k=1

M_jk.

Moreover, if|e_k|= 1 (1≤k≤m), then the equality in (3.2) holds if and only if (3.3)

n

X

j=1

kx_jk ≥ 1 m

n

X

j=1 m

X

k=1

M_jk,

and (3.4)

n

X

j=1

x_j =

n

X

j=1

kx_jk − 1 m

n

X

j=1 m

X

k=1

M_jk

! _m X

k=1

e_k.

Proof. Taking the summation in (3.1) overj from 1 ton, we obtain

n

X

j=1

kx_jk ≤Re

* e_k,

n

X

j=1

x_j +

+

n

X

j=1

M_jk,

for eachk ∈ {1, . . . , m}. Summing these inequalities overkfrom 1 tom, we deduce

(3.5)

n

X

j=1

kxjk ≤ 1 mRe

* _m X

k=1

ek,

n

X

j=1

xj

+ + 1

m

X

k=1 n

X

j=1

Mjk.

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Using the Cauchy–Schwarz we obtain

Re

* _m X

k=1

e_k,

n

X

j=1

x_j +!2

(3.6)

≤ 1 2





* _m X

k=1

ek,

n

X

j=1

xj

+

2

+

* _m X

k=1

ek,

n

X

j=1

xj

+^∗

2



≤ 1 2





m

X

k=1

e_k

2

n

X

j=1

x_j

2

+

m

X

k=1

e_k

2

n

X

j=1

x_j

2



≤

m

X

k=1

e_k

2

n

X

j=1

x_j

2

≤m

n

X

j=1

x_j

2

,

since

m

X

k=1

e_k

2

=

* _m X

k=1

e_k,

m

X

k=1

e_k +

=

m

X

k=1 m

X

l=1

he_k, e_li

=

m

X

k=1

|e_k|²

≤m .

Using (3.6) in (3.5), we deduce the desired inequality.

If (3.3) and (3.4) hold, then

√1 m

n

X

j=1

xj

= 1

√m

n

X

j=1

kxjk − 1 m

n

X

j=1 m

X

k=1

Mjk

!

m

X

k=1

ek

=

n

X

j=1

kx_jk − 1 m

n

X

j=1 m

X

k=1

M_jk,

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and the equality in (3.2) holds true.

Conversely, if the equality holds in (3.2), then obviously (3.3) is valid and we have equalities throughout the proof above. This means that

kx_jk −Rehe_k, x_ji=M_jk, Re

* _m X

k=1

e_k,

n

X

j=1

x_j +

=

* _m X

k=1

e_k,

n

X

j=1

x_j +

,

and

* _m X

k=1

e_k,

n

X

j=1

x_j +

=

m

X

k=1

e_k

n

X

j=1

x_j .

It follows from Lemma2.4and the previous relations that

n

X

j=1

x_j = Pm

k=1e_k kPm

k=1e_kk²

* _m X

k=1

e_k,

n

X

j=1

x_j +

= Pm

k=1ek

m Re

* _m X

k=1

e_k,

n

X

j=1

x_j +

= Pm

k=1e_k m

m

X

k=1 n

X

j=1

(kx_jk −M_jk)

=

n

X

j=1

kx_jk − 1 m

n

X

j=1 m

X

k=1

M_jk

! _m X

k=1

e_k.

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References

[1] A.H. ANSARI AND M.S. MOSLEHIAN, Refinements of reverse triangle in- equality in inner product spaces, J. Inequal. Pure Appl. Math., 6(3) (2005), Art. 64. [ONLINE:http://jipam.vu.edu.au/article.php?sid=

537].

[2] A.H. ANSARI AND M.S. MOSLEHIAN, More on reverse triangle inequality in inner product spaces, Inter. J. Math. Math. Sci., 18 (2005), 2883–2893.

[3] Lj. ARAMBA ˇCI ´CAND R. RAJI ´C, On theC^∗-valued triangle equality and inequality in HilbertC^∗-modules, Acta Math. Hungar., 119(4) (2008), 373–380.

[4] I. BRNETIC, S.S. DRAGOMIR, R. HOXHAANDJ. PE ˇCARI ´C, A reverse of the triangle inequality in inner product spaces and applications for polynomials, Aust. J. Math. Anal. Appl., 3(2) (2006), Art. 9.

[5] J.B. DIAZ AND F.T. METCALF, A complementary triangle inequality in Hilbert and Banach spaces, Proc. Amer. Math. Soc., 17(1) (1966), 88–97.

[6] S.S. DRAGOMIR, Reverses of the triangle inequality in inner product spaces, Linear Algebra Appl., 402 (2005), 245–254.

[7] S.S. DRAGOMIR, Some reverses of the generalized triangle inequality in com- plex inner product spaces, RGMIA Res. Rep. Coll., 7(E) (2004), Art. 7.

[8] S.S. DRAGOMIR, Reverse of the continuous triangle inequality for Bochner integral in complex Hilbert spaces, J. Math. Anal. Appl., 329 (2007), 65–76.

[9] J. KARAMATA, Teorijia i Praksa Stieltjesova Integrala (Sebro- Coratian)(Stieltjes Integral, Theory and Practice), SANU, Posebna izdanja, 154, Beograd, 1949.

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[10] M. KATO, K.S. SAITO AND T. TAMURA, Sharp triangle inequality and its reverse in Banach spaces, Math. Inequal. Appl., 10 (2007), 451–460.

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[17] M. PETROVICH, Module d’une somme, L’ Ensignement Math., 19 (1917), 53–56.