Some Slater's Type Inequalities for Convex Functions of Selfadjoint Operators in Hilbert Spaces

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FUNCTIONS OF SELFADJOINT OPERATORS IN HILBERT SPACES

S.S. DRAGOMIR

Abstract. Some inequalities of the Slater type for convex functions of selfad-joint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also pro-vided.

1. _Introduction

Suppose that I is an interval of real numbers with interior ˚I and f : I → R is a convex function onI. Then f is continuous on ˚I and has finite left and right derivatives at each point of˚I. Moreover, ifx, y∈˚I andx < y,thenf₋0 (x)≤f₊0 (x)≤ f₋0 (y)≤f+0 (y) which shows that bothf−0 andf+0 are nondecreasing function on ˚I. It is also known that a convex function must be differentiable except for at most countably many points.

For a convex function f :I →_R, the subdifferential of f denoted by∂f is the

set of all functionsϕ:I→[−∞,∞] such thatϕ˚I⊂Rand

f(x)≥f(a) + (x−a)ϕ(a) for anyx, a∈I.

It is also well known that iff is convex onI,then∂f is nonempty,f₋0 ,f₊0 ∈∂f and ifϕ∈∂f, then

f₋0 (x)≤ϕ(x)≤f+0 (x) for anyx∈˚I. In particular,ϕis a nondecreasing function.

Iff is differentiable and convex on ˚I, then∂f ={f0}.

The following result is well known in the literature asthe Slater inequality:

Theorem 1 (Slater, 1981, [15]). If f :I→Ris a nonincreasing (nondecreasing) convex function, xi ∈ I, pi ≥ 0 with Pn := P

n

i=1pi > 0 and P n

i=1piϕ(xi) 6= 0, whereϕ∈∂f, then

(1.1) 1

Pn n

X

i=1

pif(xi)≤f

Pn

i=1pixiϕ(xi)

Pn

i=1piϕ(xi)

.

Date: September 20, 2008.

1991Mathematics Subject Classification. 47A63; 47A99.

Key words and phrases. Selfadjoint operators, Positive operators, Slater’s inequality, Convex functions, Functions of selfadjoint operators.

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As pointed out in [1, p. 208], the monotonicity assumption for the derivativeϕ can be replaced with the condition

(1.2)

Pn

i=1pixiϕ(xi)

Pn

i=1piϕ(xi) ∈I,

which is more general and can hold for suitable points inI and for not necessarily monotonic functions.

2. _{Some Operator Inequalities for Convex Functions}

Let A be a selfadjoint linear operator on a complex Hilbert space (H;h., .i). The Gelfand map establishes a ∗-isometrically isomorphism Φ between the set C(Sp(A)) of allcontinuous functionsdefined on thespectrumofA,denotedSp(A), an the C∗-algebra C∗(A) generated by A and the identity operator 1H on H as follows (see for instance [7, p. 3]):

For anyf, g∈C(Sp(A)) and anyα, β∈Cwe have (i) Φ (αf+βg) =αΦ (f) +βΦ (g) ;

(ii) Φ (f g) = Φ (f) Φ (g) and Φ ¯f

= Φ (f)∗; (iii) kΦ (f)k=kfk:= sup_t_∈_Sp₍_A₎|f(t)|;

(iv) Φ (f0) = 1Hand Φ (f1) =A,wheref0(t) = 1 andf1(t) =t,fort∈Sp(A). With this notation we define

f(A) := Φ (f) for allf ∈C(Sp(A))

and we call it thecontinuous functional calculus for a selfadjoint operatorA. IfAis a selfadjoint operator andf is a real valued continuous function onSp(A), then f(t) ≥0 for any t ∈ Sp(A) implies that f(A) ≥0, i.e. f(A) is a positive operator onH.Moreover, if bothf andg are real valued functions onSp(A) then the following important property holds:

(P) f(t)≥g(t) for any t∈Sp(A) implies that f(A)≥g(A)

in the operator order ofB(H).

For a recent monograph devoted to various inequalities for functions of selfadjoint operators, see [7] and the references therein. For other results, see [14], [8] and [10]. The following result that provides an operator version for the Jensen inequality is due to Mond & Peˇcari´c [12] (see also [7, p. 5]):

Theorem 2 (Mond-Peˇcari´c, 1993, [12]). Let A be a selfadjoint operator on the Hilbert space H and assume that Sp(A) ⊆ [m, M] for some scalars m, M with m < M.If f is a convex function on[m, M],then

(MP) f(hAx, xi)≤ hf(A)x, xi,

for eachx∈H with kxk= 1.

As a special case of Theorem 2 we have the following H¨older-McCarthy inequal-ity:

Theorem 3(H¨older-McCarthy, 1967, [9]). LetAbe a selfadjoint positive operator on a Hilbert spaceH. Then

(i) hAr_{x, xi ≥ hAx, xi}r

for all r >1 andx∈H with kxk= 1; (ii) hAr_{x, xi ≤ hAx, xi}r

for all0< r <1 andx∈H withkxk= 1; (iii) If A is invertible, then hAr_{x, xi ≥ hAx, xi}r

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The following result that provides a reverse of the Mond & Peˇcari´c has been obtained in [4]:

Theorem 4 (Dragomir, 2008, [4]). LetI be an interval andf :I→_Rbe a convex and differentiable function on ˚I (the interior ofI)whose derivativef0 is continuous on ˚I.IfAis a selfadjoint operators on the Hilbert spaceHwithSp(A)⊆[m, M]⊂˚I, then

(2.1) (0≤)hf(A)x, xi −f(hAx, xi)≤ hf0₍_A₎_{Ax, xi − hAx, xi · hf}0₍_A₎_{x, xi}_,

for any x∈H with kxk= 1.

Perhaps more convenient reverses of the Mond & Peˇcari´c result are the following inequalities that have been obtained in the same paper [4]:

Theorem 5 (Dragomir, 2008, [4]). LetI be an interval andf :I→Rbe a convex and differentiable function on ˚I (the interior ofI)whose derivativef0 is continuous on ˚I.IfAis a selfadjoint operators on the Hilbert spaceHwithSp(A)⊆[m, M]⊂˚I, then

(2.2) (0≤)hf(A)x, xi −f(hAx, xi)

≤

    

   

1

2·(M−m)

h

kf0₍_A₎_xk2

− hf0₍_A₎_{x, xi}2i1/2

1 2·(f

0₍_M₎₋_f0₍_m₎₎h_kAxk2

− hAx, xi2i1/2

≤1

4(M−m) (f

0₍_M₎₋_f0₍_m₎₎_,

for any x∈H with kxk= 1. We also have the inequality

(2.3) (0≤)hf(A)x, xi −f(hAx, xi)≤1

4(M−m) (f

0₍_M₎₋_f0₍_m₎₎

−

  

 

[hM x−Ax, Ax−mxi hf0₍_M₎_x₋_f0₍_A₎_{x, f}0₍_A₎_x₋_f0₍_m₎_xi_]12_,

hAx, xi −M+₂m hf

0₍_A₎_{x, xi −}f0(M)+f0(m) 2

≤1

4(M−m) (f

0₍_M₎₋_f0₍_m₎₎_,

Moreover, if m >0 andf0(m)>0, then we also have (2.4) (0≤)hf(A)x, xi −f(hAx, xi)

≤

   

  

1 4·

(M_√−m)(f0(M)−f0(m))

M mf0₍_M₎_f0₍_m₎ hAx, xi hf

0₍_A₎_{x, xi}_,

√

M−√m pf0₍_M₎₋p

f0₍_m₎_[_{hAx, xi hf}0₍_A₎_{x, xi}_]12_, for any x∈H with kxk= 1.

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The main aim of the present paper is to provide some Slater’s type vector in-equalities for convex functions whose derivatives are continuous.

3. _{Some Slater’s Type Inequalities} The following result holds:

Theorem 6. Let I be an interval and f : I → R be a convex and differentiable function on ˚I (the interior of I) whose derivative f0 is continuous on ˚I. If A is a selfadjoint operator on the Hilbert space H with Sp(A)⊆[m, M]⊂˚I andf0(A)is a positive definite operator onH then

(3.1) 0≤f

_hAf0₍_A₎_{x, xi}

hf0₍_A₎_{x, xi}

− hf(A)x, xi

≤f0

hAf0₍_A₎_{x, xi}

hf0₍_A₎_{x, xi}

hAf0₍_A₎_{x, xi − hAx, xi hf}0₍_A₎_{x, xi}

hf0₍_A₎_{x, xi}

,

Proof. Sincef is convex and differentiable on ˚I, then we have that

(3.2) f0(s)·(t−s)≤f(t)−f(s)≤f0(t)·(t−s) for anyt, s∈[m, M].

Now, if we fixt ∈[m, M] and apply the property (P) for the operator A, then for anyx∈H withkxk= 1 we have

(3.3) hf0₍_A₎_·₍_t_·₁

H−A)x, xi ≤ h[f(t)·1H−f(A)]x, xi

≤ hf0(t)·(t·1H−A)x, xi for anyt∈[m, M] and anyx∈H withkxk= 1.

The inequality (3.3) is equivalent with

(3.4) thf0(A)x, xi − hf0(A)Ax, xi ≤f(t)− hf(A)x, xi ≤f0(t)t−f0(t)hAx, xi for anyt∈[m, M] anyx∈H withkxk= 1.

Now, since A is selfadjoint with mI ≤ A ≤ M I and f0(A) is positive defi-nite, then mf0(A)≤ Af0(A) ≤ M f0(A), i.e., mhf0₍_A₎_{x, xi ≤ hAf}0₍_A₎_{x, xi ≤}

Mhf0₍_A₎_{x, xi}_{for any}_x_∈_H _with_kxk_{= 1}_,_{which shows that}

t0:=

hf0₍_A₎_{x, xi} ∈[m, M] for anyx∈H with kxk= 1.

Finally, if we putt=t0in the equation (3.4), then we get the desired result (3.1).

Remark 1. It is important to observe that, the condition thatf0(A)is a positive definite operator onH can be replaced with the more general assumption that

(3.5) hAf

0₍_A₎_{x, xi}

hf0₍_A₎_{x, xi} ∈˚I for any x∈H with kxk= 1,

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Remark 2. Now, if the functions is concave on ˚I and the condition (3.5) holds, then we have the inequality

(3.6) 0≤ hf(A)x, xi −f

hf0₍_A₎_{x, xi}

≤f0

hf0₍_A₎_{x, xi}

hAx, xi hf0₍_A₎_{x, xi − hAf}0₍_A₎_{x, xi}

hf0₍_A₎_{x, xi}

,

The following examples are of interest:

Example 1. If Ais a positive definite operator onH, then (3.7) (0≤)hlnAx, xi −ln

A−1x, x−1

≤ hAx, xi ·

A−1x, x

−1,

Indeed, we observe that if we consider the concave function f : (0,∞) → R, f(t) = lnt,then

hf0₍_A₎_{x, xi} =

1

hA−1_{x, xi} ∈(0,∞), for anyx∈H with kxk= 1 and by the inequality (3.6) we deduce the desired result (3.7).

The following example concerning powers of operators is of interest as well:

Example 2. If A is a positive definite operator on H, then for any x∈ H with kxk= 1we have

(3.8) 0≤ hAp_{x, xi}p−1 −

Ap−1x, xp

≤phAp_{x, xi}p−2

hAp_{x, xi − hAx, xi}

Ap−1x, x

forp≥1,

(3.9) 0≤

Ap−1x, xp− hAp_{x, xi}p−1

≤phAp_{x, xi}p−2

hAx, xi

Ap−1x, x

− hAp_{x, xi}

for0< p <1, and

(3.10) 0≤ hAp_{x, xi}p−1 −

Ap−1x, xp

≤(−p)hAp_{x, xi}p−2

hAx, xi

Ap−1x, x

− hAp_{x, xi}

forp <0.

The proof follows from the inequalities (3.1) and (3.6) for the convex (concave) function f(t) = tp_{, p} _∈ ₍_−∞,₀₎_∪_[1_,_∞_{) (}_p_∈₍₀_,_{1)) by performing the required} calculation. The details are omitted.

4. _{Further Reverses}

The following results that provide perhaps more useful upper bounds for the nonnegative quantity

f

hf0₍_A₎_{x, xi}

− hf(A)x, xi forx∈H with kxk= 1,

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Theorem 7. Let I be an interval and f : I → R be a convex and differentiable function on ˚I (the interior of I) whose derivative f0 is continuous on ˚I. Assume that A is a selfadjoint operator on the Hilbert space H with Sp(A) ⊆ [m, M] ⊂˚I andf0(A)is a positive definite operator onH.If we define

B(f0, A;x) := 1 hf0₍_A₎_{x, xi}.f

0hAf0(A)x, xi

hf0₍_A₎_{x, xi}

then

(4.1) (0≤)f

hAf0(A)x, xi hf0₍_A₎_{x, xi}

− hf(A)x, xi

≤B(f0, A;x)×

    

   

1

2·(M −m)

h

kf0₍_A₎_xk2

− hf0₍_A₎_{x, xi}2i1/2

1 2·(f

0₍_M₎₋_f0₍_m₎₎h_kAxk2

− hAx, xi2i1/2

≤1

4(M−m) (f

0₍_M₎₋_f0₍_m₎₎_B₍_f0_{, A}_;_x₎

and

(4.2) (0≤)f

hf0₍_A₎_{x, xi}

− hf(A)x, xi

≤B(f0, A;x)×

₁

4(M −m) (f

0₍_M₎₋_f0₍_m₎₎

−

  

 

0₍_A₎_{x, xi −}f0₍_M₎₊_f0₍_m₎

2



 

≤ 1

4(M −m) (f

0₍_M₎₋_f0₍_m₎₎_B₍_f0_{, A}_;_x₎_,

for any x∈H with kxk= 1,respectively.

Moreover, if Ais a positive definite operator, then

(4.3) (0≤)f

hf0₍_A₎_{x, xi}

− hf(A)x, xi

≤B(f0, A;x)×

   

  

1 4 ·

(M_√−m)(f0(M)−f0(m))

0₍_A₎_{x, xi}_,

√

f0₍_m₎_[_{hAx, xi hf}0₍_A₎_{x, xi}_]12_, for any x∈H with kxk= 1.

Proof. We use the following Gr¨uss’ type result we obtained in [2]:

LetAbe a selfadjoint operator on the Hilbert space (H;h., .i) and assume that Sp(A)⊆[m, M] for some scalarsm < M.Ifhandg are continuous on [m, M] and γ:= mint∈[m,M]h(t) and Γ := maxt∈[m,M]h(t),then

(4.4) |hh(A)g(A)x, xi − hh(A)x, xi · hg(A)x, xi|

≤ 1

2 ·(Γ−γ)

h

kg(A)xk2− hg(A)x, xi2i1/2

≤ 1

4(Γ−γ) (∆−δ)

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for eachx∈Hwithkxk= 1,whereδ:= mint∈[m,M]g(t) and ∆ := maxt∈[m,M]g(t). Therefore, we can state that

(4.5) hAf0(A)x, xi − hAx, xi · hf0(A)x, xi

≤1

2 ·(M −m)

h

kf0₍_A₎_xk2

− hf0₍_A₎_{x, xi}2i1/2

≤1

4(M−m) (f

0₍_M₎₋_f0₍_m₎₎

and

(4.6) hAf0(A)x, xi − hAx, xi · hf0(A)x, xi

≤ 1 2·(f

0₍_M₎₋_f0₍_m₎₎h_kAxk2

− hAx, xi2i1/2

≤1

4(M−m) (f

0₍_M₎₋_f0₍_m₎₎_,

for eachx∈H with kxk= 1, which together with (3.1) provide the desired result (4.1).

On making use of the inequality obtained in [3]

(4.7) |hh(A)g(A)x, xi − hh(A)x, xi hg(A)x, xi| ≤ 1

4·(Γ−γ) (∆−δ)

−

  

 

[hΓx−h(A)x, f(A)x−γxi h∆x−g(A)x, g(A)x−δxi]12_,

hh(A)x, xi −

Γ+γ 2

hg(A)x, xi −∆+₂δ ,

for eachx∈H withkxk= 1,we can state that

hAf0(A)x, xi − hAx, xi · hf0(A)x, xi ≤ 1

4(M −m) (f

0₍_M₎₋_f0₍_m₎₎

−

  

 

0₍_A₎_{x, xi −}f0₍_M₎₊_f0₍_m₎

2

,

Further, in order to proof the third inequality, we make use of the following result of Gr¨uss’ type we obtained in [3]:

Ifγandδ are positive, then

(4.8) |hh(A)g(A)x, xi − hh(A)x, xi hg(A)x, xi|

≤

  

 

1 4·

(Γ−γ)(∆−δ)

√

Γγ∆δ hh(A)x, xi hg(A)x, xi,

√

Γ−√γ √∆−√δ[hh(A)x, xi hg(A)x, xi]12_,

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Now, on making use of (4.8) we can state that

hAf0(A)x, xi − hAx, xi · hf0(A)x, xi

≤

   

  

1 4·

(M_√−m)(f0(M)−f0(m))

0₍_A₎_{x, xi}_,

√

f0₍_m₎_[_{hAx, xi hf}0₍_A₎_{x, xi}_]12_,

Remark 3. We observe, from the first inequality in (4.3), that

(1≤) hAf

0₍_A₎_{x, xi}

hAx, xi hf0₍_A₎_{x, xi} ≤

1 4·

(M −m) (f0(M)−f0(m))

p

M mf0₍_M₎_f0₍_m₎ + 1

which implies that

f0

hf0₍_A₎_{x, xi}

≤f0

"

1 4·

(M−m) (f0(M)−f0(m))

p

M mf0₍_M₎_f0₍_m₎ + 1 #

hAx, xi

!

,

for eachx∈H withkxk= 1,sincef0 is monotonic nondecreasing andAis positive definite.

Now, the first inequality in (4.3) implies the following result

(4.9) (0≤)f

hf0₍_A₎_{x, xi}

− hf(A)x, xi

≤1 4 ·

(M −m) (f0(M)−f0(m))

p

M mf0₍_M₎_f0₍_m₎

×f0

"

1 4 ·

(M−m) (f0(M)−f0(m))

p

M mf0₍_M₎_f0₍_m₎ + 1 #

hAx, xi

!

hAx, xi,

From the second inequality in (4.3) we also have

(4.10) (0≤)f

hAf0(A)x, xi hf0₍_A₎_{x, xi}

− hf(A)x, xi

≤√M −√m pf0₍_M₎₋p

f0₍_m₎

×f0

"

1 4·

(M −m) (f0(M)−f0(m))

p

M mf0₍_M₎_f0₍_m₎ + 1 #

hAx, xi

!

hAx, xi hf0₍_A₎_{x, xi}

12 ,

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5. Multivariate Versions

The following result for sequences of operators can be stated.

Theorem 8. Let I be an interval and f : I → R be a convex and differentiable function on ˚I (the interior of I) whose derivativef0 is continuous on ˚I. If Aj, j∈ {1, ..., n}are selfadjoint operators on the Hilbert spaceH withSp(Aj)⊆[m, M]⊂˚I and

(5.1)

Pn

j=1hAjf0(Aj)xj, xji

Pn

j=1hf0(Aj)xj, xji ∈˚I

for eachxj∈H, j∈ {1, ..., n} withPn_j₌₁kxjk2= 1,then

(5.2) 0≤f

Pn

j=1hAjf0(Aj)xj, xji

Pn

j=1hf0(Aj)xj, xji

!

− n

X

j=1

hf(Aj)xj, xji

≤f0

Pn

j=1hAjf0(Aj)xj, xji

Pn

j=1hf0(Aj)xj, xji

!

×

" Pn

j=1hAjf0(Aj)xj, xji −P n

j=1hAjxj, xjiP n j=1hf

0₍_A

j)xj, xji

Pn

j=1hf0(Aj)xj, xji

#

,

for eachxj∈H, j∈ {1, ..., n} withP n j=1kxjk

2 = 1.

Proof. As in [7, p. 6], if we put

e

A:=



    

A1 . . . 0 .

. .

0 . . . An



    

and_ex=



    

x1 . . . xn



    

,

then we haveSpAe

⊆[m, M],k_exk= 1

D

fAe

e

x,x_eE= n

X

j=1

hf(Aj)gxj, xji,

D

f0Ae

e

x,x_eE= n

X

j=1

hf0(Aj)xj, xji,

D e

Af0Ae

e

x,x_eE= n

X

j=1

hAjf0(Aj)xj, xji

and so on.

Applying Theorem 6 under the condition (3.5) forAeandx_ewe deduce the desired

result. The details are omitted.

The following particular case is of interest

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and forpj ≥0with P n

j=1pj = 1if we also assume that

(5.3)

D Pn

j=1pjAjf0(Aj)x, x

E

D Pn

j=1pjf0(Aj)x, x

E ∈˚I

for eachx∈H with kxk= 1,then

(5.4) 0≤f



 D

Pn

j=1pjAjf0(Aj)x, x

E

D Pn

j=1pjf0(Aj)x, x

E  − * n X j=1

pjf(Aj)x, x

+

≤f0



 D

Pn

j=1pjAjf0(Aj)x, x

E

D Pn

j=1pjf0(Aj)x, x

E   ×   D Pn

j=1pjAjf0(Aj)x, x

E

−DPn

j=1pjAjx, x

E D Pn

j=1pjf0(Aj)x, x

E

D Pn

j=1pjf0(Aj)x, x

E



,

Proof. Follows from Theorem 8 on choosing xj = √

pj ·x, j ∈ {1, ..., n}, where pj ≥ 0, j ∈ {1, ..., n}, Pn_j₌₁pj = 1 and x ∈ H, with kxk = 1. The details are omitted.

The following examples are interesting in themselves:

Example 3. If Aj,j∈ {1, ..., n} are positive definite operators onH, then

(5.5) (0≤) n

X

j=1

hlnAjxj, xji −ln

     n X j=1

A−_j1xj, xj



 −1

  ≤ n X j=1

hAjxj, xji · n

X

j=1

A−_j1xj, xj−1,

for eachxj∈H, j∈ {1, ..., n} withP n j=1kxjk

2 = 1. If pj ≥0, j∈ {1, ..., n} with P

n

j=1pj= 1,then we also have the inequality

(5.6) (0≤)

* n

X

j=1

pjlnAjx, x

+ −ln      * n X j=1

pjA−j1x, x

+

 −1

  ≤ * _n X j=1

pjAjx, x

+

·

* _n X

j=1

pjA−j1x, x

+

−1,

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Example 4. If Aj, j ∈ {1, ..., n} are positive definite operators on H, then for each xj ∈H, j∈ {1, ..., n} withP

n j=1kxjk

2

= 1we have

(5.7) 0≤

  n X j=1

Ap_jxj, xj





p−1 −   n X j=1 D

Ap_j−1xj, xj

E   p ≤p   n X j=1

Ap_jxj, xj





p−2

×   n X j=1

Ap_jxj, xj

− n

X

j=1

hAjxj, xji n

X

j=1

D

Ap_j−1xj, xj

E 



forp≥1,

(5.8) 0≤

  n X j=1 D

Ap_j−1xj, xj

E   p −   n X j=1

Ap_jxj, xj





p−1

≤p   n X j=1

Ap_jxj, xj





p−2

×   n X j=1

hAjxj, xji n

X

j=1

D

Ap_j−1xj, xj

E − n X j=1

Ap_jxj, xj





for0< p <1, and

(5.9) 0≤

  n X j=1

Ap_jxj, xj





p−1 −   n X j=1 D

Ap_j−1xj, xj

E 



p

≤(−p)

  n X j=1

Ap_jxj, xj





p−2

×   n X j=1

hAjxj, xji n

X

j=1

D

Ap_j−1xj, xj

E − n X j=1

Ap_jxj, xj





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Now, for anypj≥0 withP n

j=1pj = 1and for any x∈H withkxk= 1 we also have the inequalities

(5.10) 0≤

* n

X

j=1 pjA

p jx, x

+p−1

−

* n

X

j=1 pjA

p−1 j x, x

+p

≤p

* n

X

j=1

pjApjx, x

+p−2

×   * n X j=1

pjApjx, x

+

−

* n

X

j=1

pjAjx, x

+ * n

X

j=1

pjApj−1x, x

+



forp≥1,

(5.11) 0≤

* n

X

j=1 pjA

p−1 j x, x

+p

−

* n

X

j=1 pjA

p jx, x

+p−1

≤p

* n

X

j=1

pjApjx, x

+p−2

×   * n X j=1

pjAjx, x

+ * n

X

j=1 pjA

p−1 j x, x

+

−

* n

X

j=1 pjA

p jx, x

+



for0< p <1, and

(5.12) 0≤

* n

X

j=1 pjA

p jx, x

+p−1

−

* n

X

j=1 pjA

p−1 j x, x

+p

≤(−p)

* n

X

j=1

pjApjx, x

+p−2

×   * n X j=1

pjAjx, x

+ * n

X

j=1 pjA

p−1 j x, x

+

−

* n

X

j=1 pjA

p jx, x

+



forp <0.

References

[1] S.S. Dragomir,Discrete Inequalities of the Cauchy-Bunyakovsky-Schwarz Type, Nova Science Publishers, NY, 2004.

[2] S.S. Dragomir, Gr¨uss’ type inequalities for functions of selfadjoint operators in Hilbert spaces, Preprint,RGMIA Res. Rep. Coll.,11(e) (2008), Art. 11. [ONLINE:http://www.staff.vu. edu.au/RGMIA/v11(E).asp]

[3] S.S. Dragomir, Some new Gr¨uss’ type Inequalities for functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 11(e) (2008), Art. 12. [ONLINE: http: //www.staff.vu.edu.au/RGMIA/v11(E).asp]

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[5] S.S. Dragomir, ˇCebyˇsev’s type inequalities for functions of selfadjoint operators in Hilbert spaces, PreprintRGMIA Res. Rep. Coll.,11(e) (2008), Art. 9. [ONLINE:http://www.staff. vu.edu.au/RGMIA/v11(E).asp]

[6] S.S. Dragomir and N.M. Ionescu, Some converse of Jensen’s inequality and applications.Rev. Anal. Num´er. Th´eor. Approx.23(1994), no. 1, 71–78. MR1325895 (96c:26012).

[7] T. Furuta, J. Mićić Hot, J. Peˇcarić and Y. Seo,Mond-Peˇcarić Method in Operator Inequal-ities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.

[8] A. Matković, J. Peˇcarić and I. Perić, A variant of Jensen’s inequality of Mercer’s type for operators with applications.Linear Algebra Appl.418(2006), no. 2-3, 551–564.

[9] C.A. McCarthy,cp,Israel J. Math.,5(1967), 249-271.

[10] J. Mićić, Y.Seo, S.-E. Takahasi and M. Tominaga, Inequalities of Furuta and Mond-Peˇcarić, Math. Ineq. Appl.,2(1999), 83-111.

[11] D.S. Mitrinovi´c, J.E. Peˇcari´c and A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

[12] B. Mond and J. Peˇcari´c, Convex inequalities in Hilbert space,Houston J. Math.,19(1993), 405-420.

[13] B. Mond and J. Peˇcari´c, On some operator inequalities,Indian J. Math.,35(1993), 221-232. [14] B. Mond and J. Peˇcari´c, Classical inequalities for matrix functions,Utilitas Math.,46(1994),

155-166.

[15] M.S. Slater, A companion inequality to Jensen’s inequality, J. Approx. Theory, 32(1981), 160-166.

Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia.

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