Some Jensen’s type inequalities for log-convex functions of selfad- joint operators in Hilbert spaces under suitable assumptions for the involved operators are given

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FUNCTIONS OF SELFADJOINT OPERATORS IN HILBERT SPACES

S.S. DRAGOMIR

Abstract. Some Jensen’s type inequalities for 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 pro- vided.

1. Introduction

Let A be a selfadjoint linear operator on a complex Hilbert space (H;h:; :i): The Gelfand map establishes a -isometrically isomorphism between the set C(Sp(A))of allcontinuous functionsde…ned on thespectrumofA;denotedSp(A); and the C -algebra C (A) generated byA and the identity operator1_H on H as follows (see for instance [6, p. 3]):

For anyf; g2C(Sp(A))and any ; 2Cwe have (i) ( f + g) = (f) + (g) ;

(ii) (f g) = (f) (g)and f = (f) ; (iii) k (f)k=kfk:= sup_t₂_Sp(A)jf(t)j;

(iv) (f0) = 1Hand (f1) =A;wheref0(t) = 1andf1(t) =t;fort2Sp(A): With this notation we de…ne

f(A) := (f) for allf 2C(Sp(A))

and we call it thecontinuous functional calculus for a selfadjoint operator A:

IfAis a selfadjoint operator andf is a real valued continuous function onSp(A), then f(t) 0 for any t 2 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 anyt2Sp(A) implies thatf(A) g(A) in the operator order ofB(H):

For a recent monograph devoted to various inequalities for functions of selfadjoint operators, see [6] and the references therein. For other results, see [12], [7], [11] and [9]. For recent results, see [3], [4] and [1].

Date: March 03, 2010.

1991Mathematics Subject Classi…cation. 47A63; 47A99.

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

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2. Some Jensen’s Type Inequalities for Log-convex Functions The following result that provides an operator version for the Jensen inequality for convex functions is due to Mond and Peµcari´c [10] (see also [6, p. 5]):

Theorem 1 (Mond-Peµcari´c, 1993, [10]). 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 eachx2H with kxk= 1:

Taking into account the above result and its applications for various concrete examples of convex functions, it is therefore natural to investigate the corresponding results for the case of log-convex functions, namely functions f : I ! (0;1) for whichlnf is convex.

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

for any a; b2 I and t 2[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 function. However, obviously, there are functions that are convex but not log-convex.

As an imediate consequence of the Mond-Peµcari´c inequality above we can provide the following result:

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;1) is log-convex, then

(2.1) g(hAx; xi) exphlng(A)x; xi hg(A)x; xi for eachx2H with kxk= 1:

Proof. Consider the functionf := lng; which is convex on[m; M]:Writing (MP) forf we getln [g(hAx; xi)] hlng(A)x; xi; for eachx2H withkxk = 1;which, by taking the exponential, produces the …rst inequality in (2.1).

If we also use (MP) for the exponential function, we get

exphlng(A)x; xi hexp [lng(A)]x; xi=hg(A)x; xi for eachx2H withkxk= 1and the proof is complete.

It is also important to observe that, as a special case of Theorem 1 we have the following important inequality in Operator Theory that is well known as the Hölder-McCarthy inequality:

Theorem 3(Hölder-McCarthy, 1967, [8]). LetAbe a selfadjoint positive operator on a Hilbert spaceH. Then

(i) hA^rx; xi hAx; xi^r for all r >1 andx2H with kxk= 1;

(ii) hA^rx; xi hAx; xi^r for all0< r <1 andx2H withkxk= 1;

(iii) IfAis invertible, then hA ^rx; xi hAx; xi ^r for allr >0 andx2H with kxk= 1:

Since the function g(t) = t ^r for r > 0 is log-convex, we can improve the Hölder-McCarthy inequality as follows:

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Proposition 1. LetAbe a selfadjoint positive operator on a Hilbert space H:If A is invertible, then

(2.2) hAx; xi ^r exp ln A ^r x; x A ^rx; x for allr >0 andx2H withkxk= 1:

The following reverse for the Mond-Peµcari´c inequality that generalizes the scalar Lah-Ribari´c inequality for convex functions is well known, see for instance [6, p.

57]:

Theorem 4. Let A be a selfadjoint operator on the Hilbert space H and assume thatSp(A) [m; M]for some scalarsm; M withm < M:Iff is a convex function on[m; M]; then

(2.3) hf(A)x; xi M hAx; xi

M m f(m) +hAx; xi m

M m f(M) for eachx2H with kxk= 1:

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

Theorem 5. 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;1) is log-convex, then

(2.4) hg(A)x; xi Dh [g(m)]

M1H A

M m [g(M)]

A m1H M m i

x; xE M hAx; xi

M m g(m) +hAx; xi m

M m g(M) and

(2.5) g(hAx; xi) [g(m)]

M hAx;xi

M m [g(M)]^hAx;xi

m M m

Dh[g(m)]

M1H A

M m [g(M)]

A m1H M m

ix; xE for eachx2H with kxk= 1:

Proof. Observe that, by the log-convexity ofg;we have (2.6) g(t) =g M t

M m m+ t m

M m M [g(m)]^M^M ^m^t [g(M)]^M^t ^m^m for anyt2[m; M]:

Applying the property (P) for the operatorA, we have that hg(A)x; xi h (A)x; xi

for eachx2H with kxk= 1;where (t) := [g(m)]^M^M ^m^t [g(M)]^M^t ^m^m; t2[m; M]: This proves the …rst inequality in (2.4).

Now, observe that, by the weighted arithmetic mean-geometric mean inequality we have

[g(m)]^M^M ^m^t [g(M)]^M^t ^m^m M t

M m g(m) + t m

M m g(M) for anyt2[m; M]:

Applying the property (P) for the operatorA we deduce the second inequality in (2.4).

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Further on, if we use the inequality (2.6) fort=hAx; xi 2[m; M]then we deduce the …rst part of (2.5).

Now, observe that the function introduced above can be rearranged to read as

(t) =g(m) g(M) g(m)

t m M m

; t2[m; M] showing that is a convex function on[m; M]:

Applying Mond-Peµcari´c’s inequality for we deduce the second part of (2.5) and the proof is complete.

The above result from Theorem 5 can be utilized to produce the following reverse inequality for negative powers of operators:

Proposition 2. LetAbe a selfadjoint positive operator on a Hilbert space H:If A is invertible and Sp(A) [m; M] (0< m < M);then

(2.7) A ^rx; x h

m^M1^M^H^m^AM^A^M^m1^m^Hi r

x; x M hAx; xi

M m m ^r+hAx; xi m

M m M ^r

and

(2.8) hAx; xi ^r h

g(m)^M^M^hAx;xi^m g(M)^hAx;xi^M ^m^mi ^r h

m^M1^M^H^m^AM^A^M^m1^m^Hi r

x; x for allr >0 andx2H withkxk= 1:

3. Jensen’s Inequality for Differentiable Log-convex Functions The following result provides a reverse for the Jensen type inequality (MP):

Theorem 6(Dragomir, 2008, [5]). LetJ be an interval andf :J !Rbe a convex and di¤ erentiable function on °J (the interior ofJ)whose derivativef⁰is continuous on °J:IfAis a selfadjoint operator on the Hilbert spaceH withSp(A) [m; M] °J;

then

(3.1) (0 )hf(A)x; xi f(hAx; xi) hf⁰(A)Ax; xi hAx; xi hf⁰(A)x; xi for any x2H with kxk= 1:

The following result may be stated:

Proposition 3. Let J be an interval andg:J !Rbe a di¤ erentiable log-convex function on °J whose derivativeg⁰ is continuous on °J. IfA is a selfadjoint operator on the Hilbert spaceH with Sp(A) [m; M] °J; then

(3.2) (1 )exphlng(A)x; xi g(hAx; xi)

exphD

g⁰(A) [g(A)] ¹Ax; xE

hAx; xi D

g⁰(A) [g(A)] ¹x; xEi for eachx2H with kxk= 1:

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Proof. It follows by the inequality (3.1) written for the convex function f = lng that

hlng(A)x; xi lng(hAx; xi) +D

g⁰(A) [g(A)] ¹Ax; xE

hAx; xi D

g⁰(A) [g(A)] ¹x; xE for eachx2H withkxk= 1:

Now, taking the exponential and dividing byg(hAx; xi)>0for eachx2H with kxk= 1;we deduce the desired result (3.2).

Remark 1. Let A be a selfadjoint positive operator on a Hilbert space H:If A is invertible, then

(3.3) (1 )hAx; xi^rexp ln A ^r x; x exp r hAx; xi A ¹x; x 1 for allr >0 andx2H withkxk= 1:

The following result that provides both a re…nement and a reverse of the multi- plicative version of Jensen’s inequality can be stated as well:

Theorem 7. Let J be an interval and g : J ! R be a log-convex di¤ erentiable function on °J whose derivativeg⁰ is continuous on °J. IfA is a selfadjoint operator on the Hilbert spaceH with Sp(A) [m; M] °J; then

(3.4) 1 exp g⁰(hAx; xi)

g(hAx; xi) (A hAx; xi1_H) x; x hg(A)x; xi

g(hAx; xi) D

exph

g⁰(A) [g(A)] ¹(A hAx; xi1H)i x; xE for eachx2H with kxk= 1;where1_H denotes the identity operator on H:

Proof. It is well known that if h:J !Ris a convex di¤erentiable function on °J, then the followinggradient inequality holds

h(t) h(s) h⁰(s) (t s) for anyt; s2°J.

Now, if we write this inequality for the convex functionh= lng;then we get

(3.5) lng(t) lng(s) g⁰(s)

g(s) (t s) which is equivalent with

(3.6) g(t) g(s) exp g⁰(s)

g(s) (t s) for anyt; s2°J.

Further, if we takes:=hAx; xi 2[m; M] °J;for a …xed x2H with kxk= 1;

in the inequality (3.6), then we get

g(t) g(hAx; xi) exp g⁰(hAx; xi)

g(hAx; xi) (t hAx; xi) for anyt2°J.

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Utilising the property (P) for the operatorA and the Mond-Peµcari´c inequality for the exponential function, we can state the following inequality that is of interest in itself as well:

(3.7) hg(A)y; yi g(hAx; xi) exp g⁰(hAx; xi)

g(hAx; xi) (A hAx; xi1H) y; y g(hAx; xi) exp g⁰(hAx; xi)

g(hAx; xi) (hAy; yi hAx; xi) for eachx; y2H withkxk=kyk= 1:

Further, if we put y = x in (3.7), then we deduce the …rst and the second inequality in (3.4).

Now, if we replaceswith tin (3.6) we can also write the inequality g(t) exp g⁰(t)

g(t) (s t) g(s) which is equivalent with

(3.8) g(t) g(s) exp g⁰(t)

g(t) (t s) for anyt; s2°J.

Further, if we takes:=hAx; xi 2[m; M] °J;for a …xed x2H with kxk= 1;

in the inequality (3.8), then we get

g(t) g(hAx; xi) exp g⁰(t)

g(t) (t hAx; xi) for anyt2°J.

Utilising the property (P) for the operator A, then we can state the following inequality that is of interest in itself as well:

(3.9) hg(A)y; yi g(hAx; xi)D exph

g⁰(A) [g(A)] ¹(A hAx; xi1H)i y; yE for eachx; y2H withkxk=kyk= 1:

Finally, if we puty=xin (3.9), then we deduce the last inequality in (3.4).

Remark 2. Let A be a selfadjoint positive operator on a Hilbert space H:If A is invertible, then

(3.10) 1 D exph

r 1H hAx; xi ¹A i x; xE

A ^rx; x hAx; xi^r exp r 1_H hAx; xiA ¹ x; x for allr >0 andx2H withkxk= 1:

The following reverse inequality may be proven as well:

Theorem 8. Let J be an interval and g : J ! R be a log-convex di¤ erentiable function on °J whose derivativeg⁰ is continuous on °J. IfA is a selfadjoint operator

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on the Hilbert spaceH with Sp(A) [m; M] °J; then

(3.11) (1 ) D

[g(M)]

A m1H M m [g(m)]

M1H A M m x; xE hg(A)x; xi

D

g(A) exph

(M1H A)(A m1H)

M m

g⁰(M) g(M)

g⁰(m) g(m)

i x; xE hg(A)x; xi

exp 1

4(M m) g⁰(M) g(M)

g⁰(m) g(m) for eachx2H with kxk= 1:

Proof. Utilising the inequality (3.5) we have successively

(3.12) g((1 )t+ s)

g(s) exp (1 )g⁰(s) g(s) (t s) and

(3.13) g((1 )t+ s)

g(t) exp g⁰(t) g(t) (t s) for anyt; s2°J and any 2[0;1]:

Now, if we take the power in the inequality (3.12) and the power1 in (3.13) and multiply the obtained inequalities, we deduce

(3.14) [g(t)]¹ [g(s)]

g((1 )t+ s)

exp (1 ) g⁰(t) g(t)

g⁰(s)

g(s) (t s) for anyt; s2°J and any 2[0;1]:

Further on, if we choose in (3.14)t=M; s=mand =_M^M _m^u;then, from (3.14) we get the inequality

(3.15) [g(M)]^M^u ^m^m [g(m)]^M^M ^m^u g(u)

exp (M u) (u m)

M m

g⁰(M) g(M)

g⁰(m) g(m) which, together with the inequality

(M u) (u m)

M m

1

4(M m)

produce

(3.16) [g(M)]^M^u ^m^m[g(m)]^M^M ^m^u

g(u) exp (M u) (u m)

M m

g⁰(M) g(M)

g⁰(m) g(m) g(u) exp 1

4(M m) g⁰(M) g(M)

g⁰(m) g(m) for anyu2[m; M]:

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If we apply the property (P) to the inequality (3.16) and for the operatorAwe deduce the desired result.

Remark 3. Let A be a selfadjoint positive operator on a Hilbert space H:If A is invertible and Sp(A) [m; M] (0< m < M);then

(3.17) (1 )

[g(M)]

r(^m1H A)

M m [g(m)]

r(^A ^M1H)

M m x; x hA ^rx; xi

DA ^rexph_r(M1

H A)(A m1H) M m

ix; xE

hA ^rx; xi exp

"

1

4r(M m)² mM

#

4. Applications for Ky Fan’s Inequality

Consider the functiong: (0;1)!R,g(t) = ¹_t^t ^r; r >0:Observe that for the new functionf : (0;1)!R,f(t) = lng(t)we have

f⁰(t) = r

t(1 t) andf⁰⁰(t) = 2r ¹₂ t

t²(1 t)² fort2(0;1) showing that the functiongis log-convex on the interval 0;¹₂ :

If pi >0 for i 2 f1; :::; ng with Pn

i=1pi = 1 and ti 2 0;¹₂ for i 2 f1; :::; ng; then by applying the Jensen inequality for the convex functionf (with r= 1) on the interval 0;¹₂ we get

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Pn i=1piti

1 Pn i=1piti

Yn i=1

ti

1 t_i

pi

;

which is the weighted version of the celebratedKy Fan’s inequality, see [2, p. 3].

This inequality is equivalent with Yn

i=1

1 ti

ti pi

1 Pn i=1p_it_i Pn

i=1piti

; wherepi>0fori2 f1; :::; ngwithPn

i=1pi = 1andti2 0;¹₂ fori2 f1; :::; ng: By the weighted arithmetic mean - geometric mean inequality we also have that

Xn i=1

pi(1 ti)t_i¹ Yn i=1

1 t_i ti

pi

giving the double inequality (4.2)

Xn i=1

pi(1 ti)t_i ¹ Yn i=1

(1 ti)t_i¹ ^pⁱ Xn i=1

pi(1 ti) Xn i=1

piti

! 1

: The following operator inequalities generalizing (4.2) may be stated:

Proposition 4. LetAbe a selfadjoint positive operator on a Hilbert space H:If A is invertible and Sp(A) 0;¹₂ ; then

(4.3) D

A ¹(1H A) ^rx; xE

expD

ln A ¹(1H A) ^rx; xE

h(1H A)x; xi hAx; xi ¹

r

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for eachx2H with kxk= 1 andr >0:

In particular,

(4.4) A ¹(1_H A)x; x exp ln A ¹(1_H A) x; x

h(1_H A)x; xi hAx; xi ¹ for eachx2H with kxk= 1.

The proof follows by Theorem 2 applied for the log-convex function g(t) =

1 t t

r; r >0; t2 0;¹₂ :

Proposition 5. LetAbe a selfadjoint positive operator on a Hilbert space H:If A is invertible and Sp(A) [m; M] 0;¹₂ ; then

(4.5) D

(1_H A)A ¹ ^rx; xE

*2

4 1 m m

r(^M1H A)

M m 1 M

M

r(^A ^m1H)

M m

3 5x; x

+

M hAx; xi

M m

1 m

m

r

+hAx; xi m

M m

1 M

M

r

and

(4.6) 1 hAx; xi hAx; xi

r 1 m

m

r(M hAx;xi)

M m 1 M

M

r(hAx;xi m) M m

*2

4 1 m m

r(^M1H A)

M m 1 M

M

r(^A ^m1H)

M m

3 5x; x

+

for eachx2H with kxk= 1 andr >0:

The proof follows by Theorem 5 applied for the log-convex function g(t) =

1 t t

r; r >0; t2 0;¹₂ : Finally we have:

Proposition 6. LetAbe a selfadjoint positive operator on a Hilbert space H:If A is invertible and Sp(A) 0;¹₂ ; then

(4.7) (1 )exp ln (1H A)A ¹ ^rx; x (1 hAx; xi)hAx; xi ¹ ^r

exph

r hAx; xi D

A ¹(1_H A) ¹x; xE D

(1_H A) ¹x; xE i and

(4.8) 1 D exph

r(1 hAx; xi) ¹ 1_H hAx; xi ¹A i x; xE (1H A)A ¹ ^rx; x

(1 hAx; xi)hAx; xi ¹ ^r D

exph

r(1H A) ¹ hAx; xiA ¹ 1H

i x; xE for eachx2H with kxk= 1 andr >0:

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References

[1] S. Abramovich, S. Iveli´c and J.E.Peµcari´c, Improvement of Jensen-Ste¤ensen’s inequality for superquadratic functions.Banach J. Math. Anal.4(2010), no. 1, 159–169.

[2] E.F. Beckenbach and R. Bellman,Inequalities, 4th Edition, Springer-Verlag, Berlin, 1983.

[3] S.S. Dragomir, Grüss’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].

[4] S.S. Dragomir, Some new Grüss’ 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].

[5] 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. . [ONLINE:http:

//www.staff.vu.edu.au/RGMIA/v11(E).asp].

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

[7] A. Matković, J. Peµcarić 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.

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

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

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

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

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

Mathematics, School of Engineering & Science, Victoria University, PO Box 14428, Melbourne City, MC 8001, Australia.

E-mail address: [email protected]

URL:http://www.staff.vu.edu.au/rgmia/dragomir/

School of Computational and Applied Mathematics, University of the Witwater- srand, Private Bag 3, Wits 2050, Johannesburg, South Africa