New inequalities for operator convex functions

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R E S E A R C H

Open Access

New inequalities for operator convex

functions

Vildan Bacak

*

_{and Ramazan Türkmen}

This study is a part of corresponding author’s MSc thesis.

*_{Correspondence:} [email protected] Department of Mathematics, Science Faculty, Selçuk University, Konya, Turkey

Abstract

The aim of this paper is to present some new inequalities of Hermite-Hadamard type inequalities for operator convex functions. In this paper, we use elementary

operations and give some inequalities related to the Hermite-Hadamard type. We conclude that the results given in this work are the generalization of the recent results.

MSC: 26D15; 47A63

Keywords: Hermite-Hadamard inequality; operator convex functions; self-adjoint operators

1 Introduction

Letf : [a,b]−→Rbe a convex function, then the inequality

f

a+b



≤ 

b–a b

a

f(x)dx≤f(a) +f(b)

 , a,b∈R, ()

is known in the literature as the Hermite-Hadamard inequality (see [, ] for more infor-mation).

LetXbe a vector space,x,y∈X,x=yand [x,y] ={( –t)x+ty,t∈[, ]}. We consider the functionf : [x,y]−→Rand the associated function

g(x,y) : [, ]−→R, g(x,y)(t) :=f( –t)x+ty, t∈[, ]. Note thatf is convex on [x,y] if and only ifg(x,y) is convex on [, ].

For any convex function deﬁned on a segment [x,y] ⊂ X, we have the Hermite-Hadamard integral inequality

f

x+y



≤





f( –t)x+tydt≤f(x) +f(y)

 , ()

which can be derived from the classical Hermite-Hadamard inequality () for the convex functiong(x,y) : [, ]−→R.

A real-valued continuous functionf on an intervalIis said to be operator convex (op-erator concave) if

f( –λ)A+λB≤(≥) ( –λ)f(A) +λf(B)

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in the operator order for allλ∈[, ] and for every self-adjoint operatorAandBon a Hilbert spaceH whose spectra are contained inI. Notice that a functionf is operator concave if –f is operator convex.

In recent years, many authors have been interested in giving some reﬁnements and ex-tensions of the Hermite-Hadamard inequality in (). For more about convex functions and the Hermite-Hadamard inequality, see [–].

The author in [] shows some new integral inequalities analogous to the well-known Hermite-Hadamard inequality. We give a general form of the second of these inequalities and show that the inequalities therein are satisﬁed for operator convex functions.

The author in [] shows some new Hermite-Hadamard inequalities similar to Pach-patte’s results.

Pachpatte () gives some integral inequalities analogous to the well-known Hermite-Hadamard inequality by using a fairly elementary analysis in [].

Theorem  Let f and g be real-valued,nonnegative and convex functions on[a,b].Then (i)



b–a b

a

f(x)g(x)dx≤

M(a,b) + 

N(a,b), ()

(ii)

f

a+b



g

a+b



≤ 

b–a b

a

f(x)g(x)dx+

M(a,b) + 

N(a,b), ()

where M(a,b) =f(a)g(a) +f(b)g(b),N(a,b) =f(a)g(b) +f(b)g(a).

Tunç () gives an inequality for convex functions in [] as follows.

Theorem  Let f,g: [a,b]−→Rbe two convex functions.Then

 (b–a)

b

a

(b–x)f(a)g(x) +g(a)f(x)dx

+ 

(b–a)

b

a

(x–a)f(b)g(x) +g(b)f(x)dx

≤M(a,b)

 +

N(a,b)

 +



b–a b

a

f(x)g(x)dx, ()

where M(a,b) =f(a)g(a) +f(b)g(b),N(a,b) =f(a)g(b) +f(b)g(a).

Tunç () gives another inequality for convex functions in [], too.

Theorem  Let f,g: [a,b]−→Rbe two convex functions.Then



b–a b

a

f

a+b



g(x) +g

a+b



f(x)

dx

≤ 

(b–a)

b

a

f(x)g(x)dx+ 

M(a,b) + 

N(a,b) +f

a+b



g

a+b



, ()

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Ghazanfari () gives an inequality for two operator convex functions in [] as follows.

Theorem  Let f,g:I−→Rbe operator convex functions on the interval I.Then for any self-adjoint operators A and B on a Hilbert space H with spectra in I,the inequality

f

A+B



x,x g

A+B



x,x

≤







ftA+ ( –t)Bx,x)gtA+ ( –t)Bx,x)dt

+ 

M(A,B)(x) + 

N(A,B)(x) ()

holds for any x∈H withx= ,where

M(A,B)(x) =f(A)x,xg(A)x,x+f(B)x,xg(B)x,x,

N(A,B)(x) =f(A)x,xg(B)x,x+f(B)x,xg(A)x,x.

For further inequalities, see [–].

2 Main results

In this section, we give some new Hermite-Hadamard type inequalities for operator con-vex functions and mention the diﬀerences related to the results in recent papers. We em-phasize the diﬀerence by giving an example.

The following theorem is a generalization for the product of two operator convex func-tions.

Theorem  Let f,g:I−→Rbe operator convex,nonnegative functions on the interval I.

Then for any self-adjoint operators A and B with spectra in I,we have the inequality

f

A+B



x,x g

A+B



x,x

≤







ftA+ ( –t)Bx,x)gtA+ ( –t)Bx,x)dt

+ 

k k–

i=

f(Z)x,x

g(T)x,x

+f(Z)x,x

g(T)x,x

+f(T)x,x

g(Z)x,x

+f(T)x,x

g(Z)x,x

+ 

k k–

i=

f(Z)x,x

g(Z)x,x

+f(T)x,x

g(T)x,x

+f(T)x,x

g(T)x,x

+f(Z)x,x

g(Z)x,x

, ()

where

(k–i)A+iB k =Z,

(i+ )A+ (k– (i+ ))B

k =T, ()

iA+ (k–i)B k =Z,

(k– (i+ ))A+ (i+ )B

k =T ()

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Proof Letx∈H,x=  andA,Bbe two self-adjoint operators with spectra inI. Using the convexity off,gand the change of variableu=kt, we have

f( –t)A+tBx,x= f

 –u

k

A+u

kB

x,x

= f

( –u)A+u(k– )A+B k

x,x

≤( –u)f(A)x,x+u f

(k– )A+B k

x,x

()

and

ftA+ ( –t)Bx,x= f

u kA+

 –u

k

B

x,x

= f

uA+ (k– )B

k + ( –u)B

x,x

≤u f

A+ (k– )B k

x,x

+ ( –u)f(B)x,x. ()

By the change of variableu=kt– , we have

 –u+ 

k

A+u+ 

k B

x,x

= f

( –u)(k– )A+B

k +u

(k– )A+ B k

x,x

≤( –u) f

(k– )A+B k

x,x

+u f

(k– )A+ B k

x,x

and

u+ 

k A+

 –u+ 

k

B

x,x

= f

uA+ (k– )B

k + ( –u)

A+ (k– )B k

x,x

≤u f

A+ (k– )B k

x,x

+ ( –u) f

A+ (k– )B k

x,x

.

Similarly, by using the change of variablesu=kt– ,u=kt– , . . . ,u=kt– (k– ), we have some inequalities. By the change of variableu=kt– (k– ), we get

 –u+k– 

k

A+u+k– 

k B

x,x

= f

( –u)A+ (k– )B

k +uB

x,x

≤( –u) f

A+ (k– )B k

x,x

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and

u+k– 

k A+

 –u+k– 

k

B

x,x

= f

uA+ ( –u)(k– )A+B

k x,x

≤uf(A)x,x+ ( –u) f

(k– )A+B k

x,x

.

Using the convexity off,g, we have

f

A+B



x,x

= f

tA+ ( –t)B

 +

( –t)A+tB



x,x

≤ f(tA+ ( –t)B)x,x+ f(( –t)A+tB)x,x

 ()

and

g

A+B



x,x

= g

tA+ ( –t)B

 +

( –t)A+tB



x,x

≤ g(tA+ ( –t)B)x,x+ g(( –t)A+tB)x,x

 . ()

Firstly, if we write the values obtained from the change of variableu=ktin () and (), we get

f

A+B



x,x

≤ f(tA+ ( –t)B)x,x+ f(( –t)A+tB)x,x



= f(u

A+(k–)B

k + ( –u)B)x,x+ f(( –u)A+u

(k–)A+B k )x,x

 ()

and

g

A+B



x,x

≤ g(tA+ ( –t)B)x,x+ g(( –t)A+tB)x,x



= g(u

A+(k–)B

k + ( –u)B)x,x+ g(( –u)A+u

(k–)A+B k )x,x

 . ()

If we multiply () and () and suppose ( –u)A+u(k–)_kA+B=XanduA+(k_k–)B+ ( –u)B=

Y, we get

f

A+B



x,x g

A+B



x,x

≤ 



f(X)x,x

+f(Y)x,x

g(X)x,x

+g(Y)x,x

= 

f(X)x,x

g(X)x,x

+f(X)x,x

g(Y)x,x

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+f(Y)x,x

g(X)x,x

+f(Y)x,x

g(Y)x,x

≤ 



f(X)x,x

g(X)x,x

+f(Y)x,x

g(Y)x,x

+   u f

A+ (k– )B k

x,x

+ ( –u)f(B)x,x

×

( –u)g(A)x,x+u g

(k– )A+B k

x,x

+  

( –u)f(A)x,x+u f

(k– )A+B k

x,x

×

u g

A+ (k– )B k

x,x

+ ( –u)g(B)x,x

= 

f(X)x,x

g(X)x,x

+f(Y)x,x

g(Y)x,x

+  

u( –u) f

A+ (k– )B k

x,xg(A)x,x

+u f

A+ (k– )B k

x,x g

(k– )A+B k

x,x

+ ( –u)f(B)x,xg(A)x,x

+ ( –u)uf(B)x,x g

(k– )A+B k

x,x

+  

( –u)uf(A)x,x g

A+ (k– )B k

x,x

+ ( –u)f(A)x,xg(B)x,x

+u f

(k– )A+B k

x,x g

A+ (k– )B k

x,x

+u( –u) f

(k– )A+B k

x,xg(B)x,x. ()

If we integrate both sides of inequality () over [, ], we reach

f

A+B



x,x g

A+B



x,x

≤k  /k 

ftA+ ( –t)Bx,xgtA+ ( –t)Bx,xdt +k  /k 

f( –t)A+tBx,xg( –t)A+tBx,xdt

+   f

A+ (k– )B k

x,xg(A)x,x+f(B)x,x g

(k– )A+B k

x,x

+   f

A+ (k– )B k

x,x g

(k– )A+B k

x,x

+f(B)x,xg(A)x,x

+  

f(A)x,x g

A+ (k– )B k

x,x

+ f

(k– )A+B k

x,xg(B)x,x

+  

f(A)x,xg(B)x,x+ f

(k– )A+B k

x,x g

A+ (k– )B k

x,x

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If we continue the same operations as above until the change of variableu=kt– (k– ), we have some inequalities. And then, if we sum these obtained inequalities, we get the

desired inequality.

Remark  In inequality (), if we takek= , we get the inequality in ().

Now, we show the comparison between Theorems  and  utilizing self-adjoint opera-tors (Hermitian matrices) as follows.

Example  LetA=_{ } ,B=–. _ _. Let our operator convex functions bef(X) =X_and

g(X) =X. Sincex∈Handx= , then we can choosexasx=_. From the information given above, fork= , Theorem  gives

x∗f

A+B



x

x∗g

A+B



x

≤







ftA+ ( –t)Bx,xgtA+ ( –t)Bx,xdt

+  

x∗f(A)xx∗g

A+B



x

+x∗f(B)xx∗g

A+B



x

+

x∗f

A+B



xx∗f(A)x+

x∗f

A+B



xx∗g(B)x

+  

x∗f(A)xx∗g(B)x+ 

x∗f

A+B



x

x∗g

A+B



x

+x∗f(B)xx∗g(A)x.

Putting the values of the functions in the above inequality, we get

, ≤  





ftA+ ( –t)Bx,xgtA+ ( –t)Bx,xdt+ .

⇒





ftA+ ( –t)Bx,xgtA+ ( –t)Bx,xdt≥–..

Theorem  gives

x∗f

A+B



x

x∗g

A+B



x

≤ 







+  

x∗f(A)xx∗g(A)x+x∗f(B)xx∗g(B)x

+ 

x∗f(A)xx∗g(B)x+x∗f(B)xx∗g(A)x.

Putting the values of the functions in the above inequality, we obtain

, ≤  





ftA+ ( –t)Bx,xgtA+ ( –t)Bx,xdt+ .

⇒





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So, we can conclude that our result, Theorem , is more strict than Theorem  in this case.

The following theorem is a lower bound for the product of two operator convex func-tions.

Theorem  Let f,g:I−→Rbe operator convex,nonnegative functions on the interval I.

g(A)x,x





( –t)f( –t)A+tBx,xdt

+g(B)x,x





tf( –t)A+tBx,xdt

+f(A)x,x





( –t)g( –t)A+tBx,xdt

+f(B)x,x





tg( –t)A+tBx,xdt

≤





+

M(A,B) + 

N(A,B), ()

where

M(A,B) =f(A)x,xg(A)x,x+f(B)x,xg(B)x,x,

N(A,B) =f(A)x,xg(B)x,x+f(B)x,xg(A)x,x.

Proof Letx∈H,x=  andA,Bbe two self-adjoint operators with spectra inI. Deﬁne the real-valued functionsϕx,A,B: [, ]−→Rgiven byϕx,A,B(t) = f(( –t)A+tB)x,xand

ψx,A,B: [, ]−→Rgiven byψx,A,B(t) = g(( –t)A+tB)x,x. Sincef andg are operator

convex functions, then for everyt∈[, ], we have

f( –t)A+tBx,x≤( –t)f(A)x,x+tf(B)x,x, ()

g( –t)A+tBx,x≤( –t)g(A)x,x+tg(B)x,x. () Ifa≤bandc≤dfora,b,c,d∈R, we havead+bc≤ac+bd. Using this inequality analo-gous to () and (), we get

f( –t)A+tBx,x( –t)g(A)x,x+tg(B)x,x

+g( –t)A+tBx,x( –t)f(A)x,x+tf(B)x,x

≤f( –t)A+tBx,xg( –t)A+tBx,x

+( –t)f(A)x,x+tf(B)x,x( –t)g(A)x,x+tg(B)x,x. ()

Sinceϕx,A,B(t) andψx,A,B(t) are operator convex on [, ], they are integrable on [, ] and

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inequal-ity () over [, ], we get

g(A)x,x





( –t)f( –t)A+tBx,xdt+g(B)x,x





tf( –t)A+tBx,xdt

+f(A)x,x





( –t)g( –t)A+tBx,xdt

+f(B)x,x





tg( –t)A+tBx,xdt

≤





+f(A)x,xg(A)x,x





( –t)dt

+f(A)x,xg(B)x,x+f(B)x,xg(A)x,x





t( –t)dt

+f(B)x,xg(B)x,x





tdt. ()

It can be easily controlled that





( –t)dt=





tdt= ,





t( –t)dt= .

When the above equalities are taken into account, the proof is complete.

Remark  In inequality (), if we takex= ( –t)A+tB,a=  andb= , we get the in-equality in (). Our result is more general than ().

In Theorem , we give a lower bound. But now we give both lower and upper bounds for the product of two operator convex functions.

Theorem  Let f,g:I−→Rbe operator convex,nonnegative functions on the interval I.

k–

i=

g(Z)x,x





 – (kt–i)f( –t)A+tBx,xdt

+g(T)x,x 



(kt–i)f( –t)A+tBx,xdt

+f(Z)x,x 



 – (kt–i)g( –t)A+tBx,xdt

+f(T)x,x 



(kt–i)g( –t)A+tBx,xdt

≤





≤ 

k

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+  k

k–

i=

f(Z)x,x

g(Z)x,x

+  k

k–

i=

f(Z)x,x

g(T)x,x

+f(T)x,x

g(Z)x,x

, ()

where Zand Tare deﬁned in()and()and k is the number of steps.

Proof Letx∈H,x=  andA,Bbe two self-adjoint operators with spectra inI. Using the convexity off,g and the change of variableu=kt, we have () and (). Using the analogous condition that, ifa≤bandc≤dfora,b,c,d∈R, we havead+bc≤ac+bd, we obtain

( –u) f

( –u)A+u(k– )A+B k

x,xg(A)x,x

+u f

( –u)A+u(k– )A+B k

x,x g

(k– )A+B k

x,x

+ ( –u) g

( –u)A+u(k– )A+B k

x,xf(A)x,x

+u g

( –u)A+u(k– )A+B k

x,x f

(k– )A+B k

x,x

≤ f

( –u)A+u(k– )A+B k

x,x g

( –u)A+u(k– )A+B k

x,x

+ ( –u)f(A)x,xg(A)x,x

+u f

(k– )A+B k

x,x g

(k– )A+B k

x,x

+u( –u)f(A)x,x g

(k– )A+B k

x,x

+ f

(k– )A+B k

x,xg(A)x,x. ()

If we continue the same operations as above until the change of variableu=kt– (k– ), we have some inequalities. And then, if we integrate the multiplication inequalities, we getkinequalities. These inequalities are deﬁned on [,_k), (_k,_k), . . . , (k–_k , ], respectively. The sum of the integration parts of thesekinequalities yields_ f(( –t)A+tB)x,x g(( –

t)A+tB)x,xdt. Thus, the proof is complete.

Remark  Inequality () is a general form of inequality (). Whenk=  in inequality (), we get inequality ().

Theorem  Let f,g:I−→Rbe operator convex,nonnegative functions on the interval I.

f

A+B



x,x





gtA+ ( –t)Bx,xdt

+ g

A+B



x,x





(11)

≤ f

A+B



x,x g

A+B



x,x

+  k





+ 

k k–

i=

f(Z)x,x

g(T)x,x

+f(Z)x,x

g(T)x,x

+f(T)x,x

g(Z)x,x

+f(T)x,x

g(Z)x,x

+ 

k k–

i=

f(Z)x,x

g(Z)x,x

+f(T)x,x

g(T)x,x

+f(T)x,x

g(T)x,x

+f(Z)x,x

g(Z)x,x

, ()

where Z,Z,Tand Tare deﬁned in()and()and k is the number of steps.

Proof The proof is obvious from the proofs of Theorem  and Theorem .

Remark  In Theorem , if we takek= , we get (). Theorem  is a generalization of

Theorem . If we takekas the largest number we can take in Theorem , we near the exact solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The paper is a joint work of all the authors who contributed equally to the ﬁnal version of the paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

This study was supported by the Coordinatorship of Selçuk University’s Scientiﬁc Research Project (BAP) and the Scientiﬁc and Technical Research Council of Turkey (TÜB˙ITAK). The authors would like to thank the referees for the very helpful comments and suggestions to improve this paper.

Received: 15 November 2012 Accepted: 15 January 2013 Published: 19 April 2013

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doi:10.1186/1029-242X-2013-190