New Jensen's Type Inequalities for Differentiable Log-convex Functions of Selfadjoint Operators in Hilbert Spaces

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Vol.7 (19) (2011), 67–80

NEW JENSEN’S TYPE INEQUALITIES FOR DIFFERENTIABLE LOG-CONVEX FUNCTIONS OF

SELFADJOINT OPERATORS IN HILBERT SPACES

S.S. DRAGOMIR

Abstract. Some new Jensen’s type inequalities for differentiable log- convex functions of selfadjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also provided.

1. Introduction

Jensen’s inequality for convex functions is one of the most important result in the Theory of Inequalities due to the fact that many other famous inequalities are particular cases of this for appropriate choices of the function involved, see for instance [9, p.].

The following result that provides an operator version for the Jensen inequality for convex functions is due to Mond and Peˇcari´c [10] (see also [5, p. 5]):

Theorem 1 (Mond-Peˇcari´c, 1993, [10]). Let A be a selfadjoint operator on the Hilbert spaceH and assume thatSp(A)⊆[m, M]for some scalarsm, M with m < M. If f is a convex function on[m, M], then

f(hAx, xi)≤ hf(A)x, xi (MP)

for each x∈H with kxk= 1.

Taking into account the above result and its applications for various con- crete examples of convex functions, it is therefore natural to investigate the corresponding results for the case oflog-convex functions, namely functions f :I →(0,∞) for which lnf is convex.

We observe that such functions satisfy the elementary inequality f((1−t)a+tb)≤[f(a)]^1−t[f(b)]^t

2000Mathematics Subject Classification. 47A63, 47A99.

Key words and phrases. Selfadjoint operators, positive operators, Jensen’s inequality, convex functions, functions of selfadjoint operators.

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for any a, b ∈ I and t ∈ [0,1]. Also, due to the fact that the weighted geometric mean is less than the weighted arithmetic mean, it follows that any log-convex function is a convex functions. However, obviously, there are functions that are convex but not log-convex.

As an immediate consequence of the Mond-Peˇcari´c inequality above we can provide the following result, see for instance [4]:

Theorem 2. 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 g : [m, M]→(0,∞) is log-convex, then

g(hAx, xi)≤exphlng(A)x, xi ≤ hg(A)x, xi (1.1) for each x∈H with kxk= 1.

The following reverse for the Mond-Peˇcari´c inequality that generalizes the scalar Lah-Ribari´c inequality for convex functions is well known, see for instance [5, p. 57]:

Theorem 3. Let A be a selfadjoint operator on the Hilbert space H and assume that Sp(A)⊆[m, M] for some scalarsm, M with m < M. If f is a convex function on [m, M],then

hf(A)x, xi ≤ M− hAx, xi

M−m ·f(m) +hAx, xi −m

M−m ·f(M) (1.2) for each x∈H with kxk= 1.

This result can be improved for log-convex functions as follows [4]:

Theorem 4. 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 g : [m, M]→(0,∞) is log-convex, then

hg(A)x, xi ≤

[g(m)]

M1H−A

M−m [g(M)]

A−m1H M−m

x, x

≤ M− hAx, xi

M−m ·g(m) +hAx, xi −m

M−m ·g(M) (1.3) and

g(hAx, xi)≤[g(m)]

M−hAx,xi

M−m [g(M)]

hAx,xi−m M−m

≤

[g(m)]

M1H−A

M−m [g(M)]

A−m1H M−m

x, x

(1.4) for each x∈H with kxk= 1, where 1H is the identity operator onH.

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The following result that provides both a refinement and a reverse of the multiplicative version of Jensen’s inequality for differentiable log-convex functions can be stated as well, see [4]:

Theorem 5. Let J be an interval and g : J → R be a log-convex differentiable function on ˚J whose derivative g⁰ is continuous on ˚J. If A is a selfadjoint operator on the Hilbert spaceH withSp(A)⊆[m, M]⊂ ˚J,then

1≤

exp

g⁰(hAx, xi)

g(hAx, xi) (A− hAx, xi1_H)

x, x

≤ hg(A)x, xi g(hAx, xi) ≤D

exp h

g⁰(A) [g(A)]⁻¹(A− hAx, xi1H) i

x, x E

(1.5) for each x∈H with kxk= 1.

LetAbe a selfadjoint linear operator on a complex Hilbert space (H;h., .i).

TheGelfand map establishes a ∗-isometrically isomorphism Φ between the setC(Sp(A)) of allcontinuous functions defined on the spectrum ofA,de- noted Sp(A),and the C^∗-algebra C^∗(A) generated by A and the identity operator 1H onH as follows (see for instance [5, p. 3]):

For anyf, g∈C(Sp(A)) and any α, 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) = 1H and Φ (f1) = A, where f0(t) = 1 and f1(t) = t, for t∈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.

If A is a selfadjoint operator and f is a real valued continuous function on Sp(A), then f(t) ≥ 0 for anyt ∈ Sp(A) implies that f(A) ≥0, 1_H.e.

f(A) is a positive operator on H.Moreover, if both f and gare real valued functions onSp(A) then the following important property holds:

f(t)≥g(t) for any t∈Sp(A) implies that f(A)≥g(A) (P) in the operator order of B(H).

For a recent monograph devoted to various inequalities for functions of selfadjoint operators, see [5] and the references therein. For other results, see [12], [6], [11] and [8]. For recent results, see [1] and [2].

The main aim of the present paper is to establish other Jensen’s type inequality for differentiable log-convex functions. Some applications for the logarithmic convex function g(t) = t^−r with r > 0 and t > 0 are given as well.

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2. More inequalities for differentiable log-convex functions The following results providing companion inequalities for the Jensen inequality for differentiable log-convex functions obtained in Theorem 5 hold:

Theorem 6. Let A be a selfadjoint operator on the Hilbert space H and assume thatSp(A)⊆[m, M]for some scalarsm, M withm < M.Ifg:J → (0,∞) is a differentiable log-convex function with the derivative continuous onJ˚and [m, M]⊂J˚, then

exp

hg⁰(A)Ax, xi

hg(A)x, xi −hg(A)Ax, xi

hg(A)x, xi ·hg⁰(A)x, xi hg(A)x, xi

≥

exphhg(A) lng(A)x,xi hg(A)x,xi

i

g_hg(A)Ax,xi

hg(A)x,xi

≥1 (2.1) for each x∈H with kxk= 1.

If

hg⁰(A)Ax, xi

hg⁰(A)x, xi ∈J˚for each x∈H with kxk= 1, (C) then

exp





g⁰_hg0(A)Ax,xi hg⁰(A)x,xi

g_hg0(A)Ax,xi hg⁰(A)x,xi

hg⁰(A)Ax, xi

hg⁰(A)x, xi − hAg(A)x, xi hg(A)x, xi





≥ g

_hg0(A)Ax,xi hg⁰(A)x,xi

exp

_{hg(A) ln}_g(A)x,xi

hg(A)x,xi

≥1, (2.2) for each x∈H with kxk= 1.

Proof. By the gradient inequality for the convex function lng we have g⁰(t)

g(t) (t−s)≥lng(t)−lng(s)≥ g⁰(s)

g(s) (t−s) (2.3) for any t, s∈J, which by multiplication with˚ g(t)>0 is equivalent with

g⁰(t) (t−s)≥g(t) lng(t)−g(t) lng(s)≥ g⁰(s)

g(s) (tg(t)−sg(t)) (2.4) for any t, s∈J .˚

Fixs∈J˚and apply the property (P) to get that g⁰(A)Ax, x

−s

g⁰(A)x, x

≥ hg(A) lng(A)x, xi − hg(A)x, xilng(s)

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≥ g⁰(s)

g(s) (hAg(A)x, xi −shg(A)x, xi) (2.5) for any x ∈ H with kxk = 1, which is an inequality of interest in itself as well.

Since

hg(A)Ax, xi

hg(A)x, xi ∈[m, M] for anyx∈H with kxk= 1 then on choosings:= ^hg(A)Ax,xi_hg(A)x,xi in (2.5) we get

g⁰(A)Ax, x

−hg(A)Ax, xi hg(A)x, xi

g⁰(A)x, x

≥ hg(A) lng(A)x, xi − hg(A)x, xilng

hg(A)Ax, xi hg(A)x, xi

≥0, which, by division with hg(A)x, xi>0,produces

hg⁰(A)Ax, xi

hg(A)x, xi −hg(A)Ax, xi

hg(A)x, xi ·hg⁰(A)x, xi hg(A)x, xi

≥ hg(A) lng(A)x, xi hg(A)x, xi −lng

≥0 (2.6) for any x∈H withkxk= 1.

Taking the exponential in (2.6) we deduce the desired inequality ( 2.1).

Now, assuming that the condition (C) holds, then by choosing s :=

hg⁰(A)Ax,xi

hg⁰(A)x,xi in (2.5) we get

0≥ hg(A) lng(A)x, xi − hg(A)x, xilng

hg⁰(A)Ax, xi hg⁰(A)x, xi

≥ g⁰

g

_hg₀_(A)Ax,xi

hg⁰(A)x,xi

hAg(A)x, xi −hg⁰(A)Ax, xi

hg⁰(A)x, xi hg(A)x, xi

which, by dividing with hg(A)x, xi>0 and rearranging, is equivalent with g⁰_hg0(A)Ax,xi

hg⁰(A)x,xi

g_hg0(A)Ax,xi hg⁰(A)x,xi

hg⁰(A)Ax, xi

hg⁰(A)x, xi −hAg(A)x, xi hg(A)x, xi

≥lng

−hg(A) lng(A)x, xi

hg(A)x, xi ≥0 (2.7) for any x∈H withkxk= 1.

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Finally, on taking the exponential in (2.7) we deduce the desired inequality

(2.2).

Remark 1. We observe that a sufficient condition for (C) to hold is that eitherg⁰(A) or −g⁰(A) is a positive definite operator onH.

Corollary 1. Assume that A and g are as in Theorem 6. If the condition (C)holds, then we have the double inequality

lng

≥ hg(A) lng(A)x, xi hg(A)x, xi ≥lng

, (2.8) for any x∈H withkxk= 1.

Remark 2. Assume that A is a positive definite operator on H.Since for r >0 the function g(t) =t^−r is log-convex on (0,∞) and

hg⁰(A)Ax, xi

hg⁰(A)x, xi = hA^−rx, xi hA^−r−1x, xi >0

for anyx∈Hwithkxk= 1,then on applying the inequality (2.8) we deduce the following interesting result

ln

hA^−rx, xi hA^−r−1x, xi

≤ hA^−rlnAx, xi hA^−rx, xi ≤ln

A^−r+1x, x hA^−rx, xi

!

(2.9) for any x∈H withkxk= 1.

The details of the proof are left to the interested reader.

The case of sequences of operators is embodied in the following corollary:

Corollary 2. Let Aj, j ∈ {1, . . . , n} be selfadjoint operators on the Hilbert space H and assume that Sp(A_j) ⊆ [m, M] for some scalars m, M with m < M and each j ∈ {1, . . . , n}. If g : J → (0,∞) is a differentiable log-convex function with the derivative continuous on J˚and [m, M] ⊂ J˚, then

exp

" P_n

j=1hg⁰(A_j)A_jx_j, x_ji P_n

j=1hg(A_j)x_j, x_ji

− Pn

j=1hg(Aj)Ajxj, xji Pn

j=1hg(A_j)x_j, x_ji · Pn

j=1hg⁰(Aj)xj, xji Pn

j=1hg(A_j)x_j, x_ji

#

≥ exp

h^Pⁿ

j=1hg(Aj) lng(Aj)xj,xji Pn

j=1hg(A_j)xj,xji

i

g^Pⁿ

j=1hg(A_j)Ajxj,xji Pn

j=1hg(Aj)xj,xji

≥1 (2.10) for each x_j ∈H, j ∈ {1, . . . , n} withPn

j=1kx_jk² = 1.

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If

Pn

j=1hg⁰(A_j)A_jx_j, x_ji Pn

j=1hg⁰(A_j)x_j, x_ji ∈J˚ (2.11) for each xj ∈H, j∈ {1, . . . , n} with Pn

j=1kx_jk² = 1, then exp





 g⁰^Pⁿ

j=1hg⁰(Aj)Ajxj,xji Pn

j=1hg⁰(Aj)xj,xji

g ^Pⁿ

j=1hg⁰(Aj)xj,xji

× Pn

j=1hg⁰(Aj)Ajxj, xji Pn

j=1hg⁰(A_j)x_j, x_ji − Pn

j=1hA_jg(Aj)xj, xji Pn

j=1hg(A_j)x_j, x_ji

!#

≥ g

^Pⁿ

j=1hg⁰(Aj)xj,xji

exp ^Pⁿ

j=1hg(A_j) lng(Aj)xj,xji Pn

j=1hg(A_j)xj,xji

≥1, (2.12) for each x_j ∈H, j ∈ {1, . . . , n} withP_n

j=1kx_jk² = 1.

Proof. As in [5, p. 6], if we put

Ae:=







A₁ . . . 0 .

. . 0 . . . An







and ex=





 x₁

. . . xn







then we haveSp

Ae

⊆[m, M],kxke = 1, D

g Ae

ex,xeE

=

n

X

j=1

hg(A_j)x_j, x_ji,D Aex,e exE

=

n

X

j=1

hA_jx_j, x_ji and so on.

Applying Theorem 6 for Aeand xewe deduce the desired results.

The following particular case for sequences of operators also holds:

Corollary 3. With the assumptions of Corollary 2 and if pj ≥ 0, j ∈ {1, . . . , n} withPn

j=1p_j = 1,then exp



 DPn

j=1pjg⁰(Aj)Ajx, x E DPn

j=1pjg(Aj)x, x E

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− DPn

j=1pjg(Aj)Ajx, x E DPn

j=1pjg(Aj)x, x E ·

DPn

j=1pjg⁰(Aj)x, x E DPn

j=1pjg(Aj)x, x E





≥ exp

h^Pⁿ_j=1pjg(Aj) lng(Aj)x,xi h^Pⁿj=1pjg(Aj)x,xi

g

h^Pⁿj=1pjg(Aj)Ajx,xi h^Pⁿ_j=1pjg(Aj)x,xi

≥1 (2.13)

for each x∈H, withkxk= 1.

If

DPn

j=1pjg⁰(Aj)x, x

E ∈J˚ (2.14)

for each x∈H, withkxk= 1,then

exp





 g⁰

h^Pⁿj=1pjg⁰(Aj)Ajx,xi h^Pⁿ_j=1pjg⁰(Aj)x,xi

g

h^Pⁿj=1pjg⁰(Aj)Ajx,xi h^Pⁿ_j=1pjg⁰(Aj)x,xi

×



 DPn

j=1pjg⁰(Aj)x, x E −

DPn

j=1pjAjg(Aj)x, x E DPn

j=1pjg(Aj)x, x E









≥ g

h^Pⁿ_j=1pjg⁰(Aj)Ajx,xi h^Pⁿj=1pjg⁰(Aj)x,xi

exp

h^Pⁿ_j=1pjg(Aj) lng(Aj)x,xi h^Pⁿj=1pjg(Aj)x,xi

≥1, (2.15) for each x∈H, withkxk= 1.

Proof. Follows from Corollary 2 by choosing xj = √

pj ·x, j ∈ {1, . . . , n}

wherex∈H withkxk= 1.

The following result providing different inequalities also holds:

exp

g⁰(A)

A−hg(A)Ax, xi hg(A)x, xi 1_H

x, x

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≥

*



g(A) g

_hg(A)Ax,xi

hg(A)x,xi





g(A)

x, x +

≥

* exp





g⁰_hg(A)Ax,xi

hg(A)x,xi

g_hg(A)Ax,xi

hg(A)x,xi

Ag(A)−hg(A)Ax, xi hg(A)x, xi g(A)



x, x +

If the condition (C)from Theorem 6 holds, then

* exp



 g⁰

g

_hg₀_(A)Ax,xi

hg⁰(A)x,xi

hg⁰(A)Ax, xi

hg⁰(A)x, xi g(A)−Ag(A)



x, x +

≥

* g

[g(A)]⁻¹ g(A)

x, x +

≥

exp

g⁰(A)

hg⁰(A)Ax, xi

hg⁰(A)x, xi 1_H −A

x, x

Proof. By taking the exponential in (2.4) we have the following inequality exp

g⁰(t) (t−s)

≥ g(t)

g(s) g(t)

≥exp g⁰(s)

g(s) (tg(t)−sg(t))

(2.18) for any t, s∈J .˚

If we fix s ∈ J˚and apply the property (P) to the inequality (2.18), we deduce

exp

g⁰(A) (A−s1H) x, x

≥

* g(A)

g(s) g(A)

x, x +

≥

exp g⁰(s)

g(s) (Ag(A)−sg(A))

x, x

(2.19) for each x∈H withkxk= 1,where 1_H is the identity operator onH.

By Mond-Peˇcari´c’s inequality applied for the convex function exp we also have

exp

g⁰(s)

g(s) (Ag(A)−sg(A))

x, x

≥exp

g⁰(s)

g(s) (hAg(A)x, xi −shg(A)x, xi)

(2.20)

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for each s∈J˚andx∈H withkxk= 1.

Now, if we chooses:= ^hg(A)Ax,xi_hg(A)x,xi ∈[m, M] in (2.19) and (2.20) we deduce the desired result (2.16).

Observe that, the inequality (2.18) is equivalent with exp

g⁰(s)

g(s) (sg(t)−tg(t))

≥ g(s)

g(t) g(t)

≥exp

g⁰(t) (s−t)

(2.21) for any t, s∈J .˚

If we fix s ∈ J˚and apply the property (P) to the inequality (2.21) we deduce

exp

g⁰(s)

g(s) (sg(A)−Ag(A))

x, x

≥

g(s) [g(A)]⁻¹g(A)

x, x

≥ exp

g⁰(A) (s1_H −A) x, x

(2.22) for each x∈H withkxk= 1.

By Mond-Peˇcari´c’s inequality we also have exp

g⁰(A) (s1_H −A) x, x

≥exp s

g⁰(A)x, x

−

g⁰(A)Ax, x

(2.23) for each s∈J˚andx∈H withkxk= 1.

Taking into account that the condition (C) is valid, then we can choose in (2.22) and (2.23)s:= ^hg_hg⁰^(A)Ax,xi0(A)x,xi to get the desired result (2.17).

Remark 3. If we apply, for instance, the inequality (2.16) for the log- convex functiong(t) =t⁻¹, t >0,then, after simple calculations, we get the inequality

*

exp A⁻²−

A⁻¹x, x A⁻¹ A⁻²− hA⁻¹x, xi

! x, x

+

≥D

A⁻¹x, x

A⁻¹A⁻¹

x, xE

≥

*

exp A⁻¹−

A⁻¹x, x 1_H hA⁻¹x, xi²

! x, x

+

≥1 (2.24) for each x∈H withkxk= 1.

Other similar results can be obtained from the inequality (2.17), however the details are left to the interested reader.

3. A reverse inequality

The following reverse inequality that provides a companion for the results in Theorem 4 is also of interest:

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(1≤)[g(m)]

M−hAx,xi

M−m [g(M)]

hAx,xi−m M−m

exphlng(A)x, xi

≤exp

h(M1_H −A) (A−m1_H)x, xi M−m

g⁰(M)

g(M) −g⁰(m) g(m)

≤exp

(M− hAx, xi) (hAx, xi −m) M−m

g⁰(M)

g(M) −g⁰(m) g(m)

≤exp 1

4(M −m)

g⁰(M)

g(M) − g⁰(m) g(m)

Proof. Utilizing the inequality (2.3) we have successively lng((1−λ)t+λs)−lng(s)≥(1−λ)g⁰(s)

g(s) (t−s) (3.2) and

lng((1−λ)t+λs)−lng(t)≥ −λg⁰(t)

g(t) (t−s) (3.3) for any t, s∈˚J and any λ∈[0,1].

Now, if we multiply (3.2) by λand (3.3) by 1−λand sum the obtained inequalities, we deduce

(1−λ) lng(t) +λlng(s)−lng((1−λ)t+λs)

≤(1−λ)λ

g⁰(t)

g(t) −g⁰(s) g(s)

(t−s)

(3.4) for any t, s∈˚J and any λ∈[0,1].

Now, if we chooseλ:= _M−m^M−u, s:=mandt:=M in (3.4) then we get the inequality

u−m

M−mlng(M) + M −u

M−mlng(m)−lng(u)

≤

(M −u) (u−m) M −m

g⁰(M)

g(M) − g⁰(m) g(m)

(3.5) for any u∈[m, M].

If we use the property (P) for the operatorA we get

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hAx, xi −m

M −m lng(M) +M − hAx, xi

M−m lng(m)− hlng(A)x, xi

≤

h(M1H−A) (A−m1H)x, xi M −m

g⁰(M)

g(M) − g⁰(m) g(m)

(3.6) for each x∈H withkxk= 1.

Taking the exponential in (3.6) we deduce the first inequality in ( 3.1).

Now, consider the functionh: [m, M]→R,h(t) = (M−t) (t−m).This function is concave in [m, M] and by Mond-Peˇcari´c’s inequality ( MP) we have

h(M1_H −A) (A−m1_H)x, xi ≤(M− hAx, xi) (hAx, xi −m) for each x∈H withkxk= 1,which proves the second inequality in (3.1).

For the last inequality, we observe that (M− hAx, xi) (hAx, xi −m)≤ 1

4(M−m)²,

and the proof is complete.

Corollary 4. Assume that g is as in Theorem 8 and Aj are selfadjoint operators with Sp(Aj)⊆[m, M]⊂˚J, j∈ {1, . . . , n}.

If andx_j ∈H, j ∈ {1, . . . , n} withPn

j=1kx_jk² = 1, then (1≤)[g(m)]

M−Pn

j=1hAj xj ,xji

M−m [g(M)]

Pn

j=1hAj xj ,xji^−m

M−m

exp Pn

j hlng(Aj)xj, xji

≤exp

" Pn

j=1h(M1H −Aj) (Aj−m1H)xj, xji M−m

g⁰(M)

g(M) −g⁰(m) g(m)

#

≤exp

"

M−Pn

j=1hA_jxj, xji Pn

j=1hA_jxj, xji −m M−m

×

g⁰(M)

g(M) −g⁰(m) g(m)

#

≤exp 1

4(M−m)

g⁰(M)

g(M) −g⁰(m) g(m)

. (3.7) If p_j ≥0, j ∈ {1, . . . , n} with P_n

j=1p_j = 1, then (1≤)[g(m)]

M−h^Pnj=1pj Aj x,xi

M−m [g(M)]

h^Pnj=1pj Aj x,xi^−m

M−m

DQn

j=1[g(A_j)]^p^jx, xE

≤exp

" Pn

j=1pjh(M1H −Aj) (Aj−m1H)xj, xji M−m

g⁰(M)

g(M) −g⁰(m) g(m)

#

(13)

≤exp

"

M−D Pn

j=1pjAjx, x

E DPn

j=1pjAjx, x E

−m M−m

×

g⁰(M)

g(M) −g⁰(m) g(m)

#

≤exp 1

4(M −m)

g⁰(M)

g(M) − g⁰(m) g(m)

Remark 4. LetA be a selfadjoint positive operator on a Hilbert spaceH.

IfA is invertible, then (1≤)m

hAx,xi−M M−m M

m−hAx,xi M−m

exphlnA⁻¹x, xi ≤exp

h(M1_H−A) (A−m1_H)x, xi M m

≤exp

(M − hAx, xi) (hAx, xi −m) M m

≤exp

"

1 4

(M−m)² mM

# (3.9) for all x∈H withkxk= 1.

References

[1] 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].

[2] 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].

[3] S. S. Dragomir, Some reverses of the Jensen inequality for functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 11 (e) (2008), Art.15 [ONLINE: http://www.staff.vu.edu.au/RGMIA/v11(E).asp].

[4] S. S. Dragomir,Some Jensen type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces, Preprint RGMIA Res. Rep. Coll., 13(e) (2010), Art. 2.

[ONLINE: http://rgmia.org/v13(E).php].

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

[6] 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 (2–3) (2006), 551–564.

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

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

[9] D. S. Mitrinovi´c, J. Peˇcari´c and A. M. Fink,Classical and New Inequalities in Anal- ysis, Kluwer Academic Publishers, Dordrecht, 1993.

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

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[11] B. Mond and J. Peˇcari´c,On some operator inequalities, Indian J. Math., 35 (1993), 221–232.

[12] B. Mond and J. Peˇcari´c, Classical inequalities for matrix functions, Utilitas Math., 46 (1994), 155–166.

(Received: July 28, 2010) Mathematics, School of Engineering & Science Victoria University, PO Box 14428

Melbourne City, MC 8001, Australia E–mail: [email protected] URL:http://www.staff.vu.edu.au/rgmia/dragomir/