Mi´ci´c et al. Journal of Inequalities and Applications 2013, 2013:353 http://www.journalofinequalitiesandapplications.com/content/2013/1/353

RESEARCH

Open Access

Refined converses of Jensen’s inequality for operators Jadranka Mi´ci´c1* , Josip Peˇcari´c2 and Jurica Peri´c3 *

Correspondence: [email protected] Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Luˇci´ca 5, Zagreb, 10000, Croatia Full list of author information is available at the end of the article 1

Abstract In this paper converses of a generalized Jensen’s inequality for a continuous field of self-adjoint operators, a unital field of positive linear mappings and real-valued continuous convex functions are studied. New refined converses are presented by using the Mond-Peˇcari´c method improvement. Obtained results are applied to refine selected inequalities with power functions. MSC: 47A63; 47A64 Keywords: Mond-Peˇcari´c method; self-adjoint operator; positive linear mapping; converse of Jensen’s operator inequality; convex function

1 Introduction Let T be a locally compact Hausdorff space and let A be a C ∗ -algebra of operators on some Hilbert space H. We say that a field (xt )t∈T of operators in A is continuous if the function t → xt is norm continuous on T. If in addition μ is a Radon measure on T and  the function t → xt  is integrable, then we can form the Bochner integral T xt dμ(t), which is the unique element in A such that   xt dμ(t) = ϕ(xt ) dμ(t)

 ϕ T

T

for every linear functional ϕ in the norm dual A∗ . Assume further that there is a field (φt )t∈T of positive linear mappings φt : A → B from A to another C ∗ -algebra B of operators on a Hilbert space K . We recall that a linear mapping φ : A → B is said to be positive if φ(x) ≥  for all x ≥ . We say that such a field (φt )t∈T is continuous if the function t → φt (x) is continuous for every x ∈ A. Let the C ∗ -algebras include the identity operators and let the function t → φt (H ) be integrable   with T φt (H ) dμ(t) = kK for some positive scalar k. If T φt (H ) dμ(t) = K , we say that a field (φt )t∈T is unital. Let B(H) be the C ∗ -algebra of all bounded linear operators on a Hilbert space H. We define bounds of a self-adjoint operator x ∈ B(H) by mx := inf xξ , ξ and ξ =

Mx := sup xξ , ξ ξ =

()

for ξ ∈ H. If Sp(x) denotes the spectrum of x, then Sp(x) ⊆ [mx , Mx ]. © 2013 Mi´ci´c et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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For an operator x ∈ B(H), we define the operator |x| := (x∗ x)/ . Obviously, if x is selfadjoint, then |x| = (x )/ . Jensen’s inequality is one of the most important inequalities. It has many applications in mathematics and statistics and some other well-known inequalities are its special cases. Let f be an operator convex function defined on an interval I. Davis [] proved the socalled Jensen operator inequality     f φ(x) ≤ φ f (x) ,

()

where φ : A → B(K) is a unital completely positive linear mapping from a C ∗ -algebra A to linear operators on a Hilbert space K , and x is a self-adjoint element in A with spectrum in I. Subsequently, Choi [] noted that it is enough to assume that φ is unital and positive. Mond, Pečarić, Hansen, Pedersen et al. in [–] studied another generalization of () for operator convex functions. Moreover, Hansen et al. [] presented a general formulation of Jensen’s operator inequality for a bounded continuous field of self-adjoint operators and a unital field of positive linear mappings:  f T

    φt (xt ) dμ(t) ≤ φt f (xt ) dμ(t),

()

T

where f is an operator convex function. There is an extensive literature devoted to Jensen’s inequality concerning different refinements and extensive results, e.g., see [–]. Mićić et al. [] proved that the discrete version of () stands without operator convexity of f under a condition on the spectra of operators. Recently, Mićić et al. [] presented a discrete version of refined Jensen’s inequality for real-valued continuous convex functions. A continuous version is given below. Theorem  Let (xt )t∈T be a bounded continuous field of self-adjoint elements in a unital C ∗ -algebra A defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ. Let mt and Mt , mt ≤ Mt , be the bounds of xt , t ∈ T. Let (φt )t∈T be a unital field of positive linear mappings φt : A → B from A to another unital C ∗ -algebra B . Let (mx , Mx ) ∩ [mt , Mt ] = ∅,

t ∈ T, and a < b,

where mx and Mx , mx ≤ Mx , are the bounds of the operator x = a = sup{Mt : Mt ≤ mx , t ∈ T},

 T

φt (xt ) dμ(t) and

b = inf{mt : mt ≥ Mx , t ∈ T}.

If f : I → R is a continuous convex (resp. concave) function provided that the interval I contains all mt , Mt , then  f T

       φt (xt ) dμ(t) ≤ φt f (xt ) dμ(t) – δf x¯ ≤ φt f (xt ) dμ(t) T

T

(resp.  f T

       φt (xt ) dμ(t) ≥ φt f (xt ) dμ(t) – δf x¯ ≥ φt f (xt ) dμ(t)) T

T

()

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holds, where   ¯ ¯ +M m ¯ ¯ ¯ M) = f (m) ¯ + f (M) – f δf ≡ δf (m,      ¯ ¯ ¯ , ¯ = f m + M – f (m) ¯ – f (M) ¯ M) resp. δf ≡ δf (m,    ¯   ¯ ¯ =  K –  x – m + M K  ¯ M) x¯ ≡ x¯ x (m,  ¯ –m   ¯ M ¯ ∈ [Mx , b], m ¯ are arbitrary numbers. ¯ ∈ [a, mx ], M ¯ < M, and m The proof is similar to [, Theorem ] and we omit it. On the other hand, Mond, Pečarić, Furuta et al. in [, –] investigated converses of Jensen’s inequality. For presenting these results, we introduce some abbreviations. Let f : [m, M] → R, m < M. Then a linear function through (m, f (m)) and (M, f (M)) has the form h(z) = kf z + lf , where

kf :=

f (M) – f (m) M–m

and

lf :=

Mf (m) – mf (M) . M–m

()

Using the Mond-Pečarić method, in [] the following generalized converse of Jensen’s operator inequality () is presented

   

F φ f (A) , g φ(A) ≤ max F kf z + lf , g(z) n˜ , m≤z≤M

()

for a convex function f defined on an interval [m, M], m < M, where g is a real-valued continuous function on [m, M], F(u, v) is a real-valued function defined on U ×V , operator monotone in u, U ⊃ f [m, M], V ⊃ g[m, M], φ : Hn → Hn˜ is a unital positive linear mapping and A is a self-adjoint operator with spectrum contained in [m, M].  A continuous version of () and in the case of T φt (H ) dμ(t) = kK for some positive scalar k, is presented in []. Recently, Mićić et al. [] obtained better bound than the one given in () as follows. Theorem  [, Theorem .] Let (xt )t∈T be a bounded continuous field of self-adjoint elements in a unital C ∗ -algebra A with the spectra in [m, M], m < M, defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ, and let (φt )t∈T be a unital field of positive linear maps φt : A → B from A to another unital C ∗ -algebra B .  Let mx and Mx , mx ≤ Mx , be the bounds of the self-adjoint operator x = T φt (xt ) dμ(t) and f : [m, M] → R, g : [mx , Mx ] → R, F : U × V → R, where f ([m, M]) ⊆ U, g([mx , Mx ]) ⊆ V and F is bounded. If f is convex and F is an operator monotone in the first variable, then 

  φt f (xt ) dμ(t), g

F T



 φt (xt ) dμ(t) T

≤ C K ≤ CK ,

()

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where constants C ≡ C (F, f , g, m, M, mx , Mx ) and C ≡ C(F, f , g, m, M) are C =

sup

mx ≤z≤Mx

=



F kf z + lf , g(z)

sup

 

F pf (m) + ( – p)f (M), g pm + ( – p)M ,

M–Mx M–mx M–m ≤p≤ M–m



C = sup F kf z + lf , g(z) m≤z≤M

 

= sup F pf (m) + ( – p)f (M), g pm + ( – p)M . ≤p≤

If f is concave, then reverse inequalities are valid in () with inf instead of sup in bounds C and C. In this paper, we present refined converses of Jensen’s operator inequality. Applying these results, we further refine selected inequalities with power functions.

2 Main results In the following we assume that (xt )t∈T is a bounded continuous field of self-adjoint elements in a unital C ∗ -algebra A with the spectra in [m, M], m < M, defined on a locally compact Hausdorff space T equipped with a bounded Radon measure μ and that (φt )t∈T is a unital field of positive linear mappings φt : A → B between C ∗ -algebras. For convenience, we introduce abbreviations x and δf as follows:  

x ≡ xxt ,φt (m, M) := K –  M–m

   m + M   φt xt – H  dμ(t),  T



()

where m, M, m < M, are some scalars such that the spectra of xt , t ∈ T, are in [m, M];  m+M , δf ≡ δf (m, M) := f (m) + f (M) – f  

()

where f : [m, M] → R is a continuous function. H ≤ xt – m+M H ≤ M–m H for t ∈ T and Obviously, mH ≤ xt ≤ MH implies – M–m      m+M M–m M–m K . It follows x ≥ . Also, if f is T φt (|xt –  H |) dμ(t) ≤ T φt (H ) dμ(t) =   convex (resp. concave), then δf ≥  (resp. δf ≤ ). To prove our main result related to converse Jensen’s inequality, we need the following lemma. Lemma  Let f be a convex function on an interval I, m, M ∈ I and p , p ∈ [, ] such that p + p = . Then    m+M min{p , p } f (m) + f (M) – f  ≤ p f (m) + p f (M) – f (p m + p M)    m+M . ≤ max{p , p } f (m) + f (M) – f 

()

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Proof These results follow from [, Theorem , p.] for n = . For the reader’s convenience, we give an elementary proof of (). Let ai ≤ bi , i = , , be positive real numbers such that A = a + a < B = b + b . Using Jensen’s inequality and its reverse, we get    a m + a M b m + b M – Af Bf B A   (b – a )m + (b – a )M ≤ (B – A)f B–A 

≤ (b – a )f (m) + (b – a )f (M)   = b f (m) + b f (M) – a f (m) + a f (M) .

()

Suppose that  < p < p < , p + p = . Replacing a and a by p and p , respectively, and putting b = b = p , A =  and B = p in (), we get  p f

m+M 



    – f p f (m) + p f (M) ≤ p f (m) + p f (M) – p f (m) + p f (M) ,

which gives the right-hand side of (). Similarly, replacing b and b by p and p , respectively, and putting a = a = p , A = p and B =  in (), we obtain the left-hand side of (). If p = , p =  or p = , p = , then inequality () holds, since f is convex. If p = p = /, then we have an equality in ().  The main result of an improvement of the Mond-Pečarić method follows. Lemma  Let (xt )t∈T , (φt )t∈T , m and M be as above. Then 







 φt (xt ) dμ(t) + lf K – δf x ≤ kf

φt f (xt ) dμ(t) ≤ kf T

T

φt (xt ) dμ(t) + lf K

()

T

for every continuous convex function f : [m, M] → R, where x and δf are defined by () and (), respectively. If f is concave, then the reverse inequality is valid in (). Proof We prove only the convex case. By using () we get    m+M f (p m + p M) ≤ p f (m) + p f (M) – min{p , p } f (m) + f (M) – f 

()

for every p , p ∈ [, ] such that p + p = . Let functions p , p : [m, M] → [, ] be defined by p (z) =

M–z , M–m

p (z) =

z–m . M–m

Then, for any z ∈ [m, M], we can write  f (z) = f

   M–z z–m m+ M = f p (z)m + p (z)M . M–m M–m

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By using () we get    m+M z–m M–z f (m) + f (M) – z˜ f (m) + f (M) – f , f (z) ≤ M–m M–m 

()

where    m + M    z˜ = – z– ,  M – m   since     m + M     M–z z–m z – . , = – min M–m M–m  M – m   Now since Sp(xt ) ⊆ [m, M], by utilizing the functional calculus to (), we obtain    xt – m M – xt m+M f (m) + f (M) – xt f (m) + f (M) – f , f (xt ) ≤ M–m M–m  where      m + M 

xt – H . x t = H –  M – m  Applying a positive linear mapping φt , integrating and using the first inequality in () since 

 

x = φt ( xt ) dμ(t) = K –  M–m T

 T

φt (H ) dμ(t) = K , we get

   m + M   H  dμ(t). φt  x t –  T



x ≥ , the second inequality in () holds. By using that δf



We can use Lemma  to obtain refinements of some other inequalities mentioned in the introduction. First, we present a refinement of Theorem .  Theorem  Let mx and Mx , mx ≤ Mx , be the bounds of the operator x = T φt (xt ) dμ(t) x. Let f : [m, M] → R, g : [mx , Mx ] → R, and let m x be the lower bound of the operator F : U × V → R, where f ([m, M]) ⊆ U, g([mx , Mx ]) ⊆ V and F is bounded. If f is convex and F is operator monotone in the first variable, then 

  φt f (xt ) dμ(t), g

F T



 φt (xt ) dμ(t) T







x, g ≤ F kf x + lf – δf

φt (xt ) dμ(t) T

≤

sup

mx ≤z≤Mx



F kf z + lf – δf m x , g(z) K ≤

sup

mx ≤z≤Mx



F kf z + lf , g(z) K .

If f is concave, then the reverse inequality is valid in () with inf instead of sup.

()

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Proof We prove only the convex case. Then δf ≥  implies  ≤ δf m x K ≤ δf x. By using () it follows that 

  φt f (xt ) dμ(t) ≤ kf x + lf – δf x ≤ kf x + lf – δf m x K ≤ kf x + lf .

T



Using operator monotonicity of F(·, v) in the first variable, we obtain ().

3 Difference-type converse inequalities By using Jensen’s operator inequality, we obtain that  αg

    φt (xt ) dμ(t) ≤ φt f (xt ) dμ(t)

T

()

T

holds for every operator convex function f on [m, M], every function g and real number α such that αg ≤ f on [m, M]. Now, applying Theorem  to the function F(u, v) = u – αv, α ∈ R, we obtain the following converse of (). It is also a refinement of [, Theorem .]. Theorem  Let mx and Mx , mx ≤ Mx , be the bounds of the operator x = and f : [m, M] → R, g : [mx , Mx ] → R be continuous functions. If f is convex and α ∈ R, then 

  φt f (xt ) dμ(t) – αg

T



 T

φt (xt ) dμ(t)

   φt (xt ) dμ(t) ≤ max kf z + lf – αg(z) K – δf x. mx ≤z≤Mx

T

()

If f is concave, then the reverse inequality is valid in () with min instead of max. Remark  () Obviously, 







φt f (xt ) dμ(t) – αg T

 φt (xt ) dμ(t)

T

≤ max

mx ≤z≤Mx

    kf z + lf – αg(z) K – δf y ≤ max kf z + lf – αg(z) K mx ≤z≤Mx

for every convex function f , every α ∈ R, and m x K ≤ y ≤ x, where m x is the lower bound of x. () According to [, Corollary .], we can determine the constant in the RHS of (). (i) Let f be convex. We can determine the value Cα in  T

  φt f (xt ) dμ(t) – αg



 φt (xt ) dμ(t) ≤ Cα K – δf x

T

as follows: • if α ≤ , g is convex or α ≥ , g is concave, then   Cα = max kf mx + lf – αg(mx ), kf Mx + lf – αg(Mx ) ;

()

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• if α ≤ , g is concave or α ≥ , g is convex, then

Cα =

⎧ ⎪ kf mx + lf – αg(mx ) ⎪ ⎪ ⎪ ⎪ ⎨k z + l – αg(z ) f 

if αg– (z) ≥ kf for every z ∈ (mx , Mx ), if αg– (z ) ≤ kf ≤ αg+ (z )



f

()

⎪ ⎪ for some z ∈ (mx , Mx ), ⎪ ⎪ ⎪ ⎩ kf Mx + lf – αg(Mx ) if αg+ (z) ≤ kf for every z ∈ (mx , Mx ).

(ii) Let f be concave. We can determine the value cα in  cα K – δf x≤

  φt f (xt ) dμ(t) – αg



 φt (xt ) dμ(t)

T

T

as follows: • if α ≤ , g is convex or α ≥ , g is concave, then cα is equal to the right-hand side in () with reverse inequality signs; • if α ≤ , g is concave or α ≥ , g is convex, then cα is equal to the right-hand side in () with min instead of max. Theorem  and Remark () applied to functions f (z) = zp and g(z) = zq give the following corollary, which is a refinement of [, Corollary .]. Corollary  Let (xt )t∈T be a field of strictly positive operators, let mx and Mx , mx ≤ Mx , be  x be defined by (). the bounds of the operator x = T φt (xt ) dμ(t). Let (i) Let p ∈ (–∞, ] ∪ [, ∞). Then 

 p φt xt dμ(t) – α



q φt (xt ) dμ(t)

T

T

  x, ≤ Cα K – mp + Mp – –p (m + M)p

where the constant Cα is determined as follows: • if α ≤ , q ∈ (–∞, ] ∪ [, ∞) or α ≥ , q ∈ (, ), then   Cα = max ktp mx + ltp – αmqx , ktp Mx + ltp – αMxq ;

()

• if α ≤ , q ∈ (, ) or α ≥ , q ∈ (–∞, ] ∪ [, ∞), then ⎧ q ⎪ p p ⎪ ⎨kt mx + lt – αmx

if (αq/ktp )/(–q) ≤ mx ,

Cα = ltp + α(q – )(αq/ktp )q/(–q) ⎪ ⎪ ⎩ q ktp Mx + ltp – αMx

if mx ≤ (αq/ktp )/(–q) ≤ Mx , if

(αq/ktp )/(–q)

()

≥ Mx ,

where ktp := (Mp – mp )/(M – m) and ltp := (Mmp – mMp )/(M – m). (ii) Let p ∈ (, ). Then cα K



+ 

–p

 x≤ (m + M) – m – M p

p



p

 p φt xt dμ(t) – α

T

where the constant cα is determined as follows:



q φt (xt ) dμ(t) T

,

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• if α ≤ , q ∈ (–∞, ] ∪ [, ∞) or α ≥ , q ∈ (, ), then cα is equal to the right-hand side in (); • if α ≤ , q ∈ (, ) or α ≥ , q ∈ (–∞, ] ∪ [, ∞), then cα is equal to the right-hand side in () with min instead of max. Using Theorem  and Remark  for g ≡ f and α =  and utilizing elementary calculations, we obtain the following converse of Jensen’s inequality. Theorem  Let mx and Mx , mx ≤ Mx , be the bounds of the operator x = and let f : [m, M] → R be a continuous function. If f is convex, then 

  φt f (xt ) dμ(t) – f

≤ T



 T

φt (xt ) dμ(t)



¯ K – δf x, φt (xt ) dμ(t) ≤ C

()

T

where x and δf are defined by () and (), respectively, and C¯ = max

mx ≤z≤Mx

  kf z + lf – f (z) .

()

¯ K – δf Furthermore, if f is strictly convex differentiable, then the bound C x satisfies the following condition:   ¯ K – δf  ≤ C x ≤ f (M) – f (m) – f  (m)(M – m) – δf m x K , where m x is the lower bound of the operator x. We can determine the value C¯ in () as follows: C¯ = kf z + lf – f (z ),

()

where ⎧ ⎪ ⎪ ⎨ mx z =

⎪ ⎪ ⎩

f – (kf )

Mx

if f  (mx ) ≥ kf , if f  (mx ) ≤ kf ≤ f  (Mx ),

()



if f (Mx ) ≤ kf .

In the dual case, when f is concave, the reverse inequality is valid in () with min instead ¯ K – δf x of max in (). Furthermore, if f is strictly concave differentiable, then the bound C satisfies the following condition:   ¯ K – δf f (M) – f (m) – f  (m)(M – m) – δf m x K ≤ C x ≤ . We can determine the value C¯ in () with z , which equals the right-hand side in () with reverse inequality signs. Example  We give examples for the matrix cases and T = {, }. We put f (t) = t  , which is convex, but not operator convex. Also, we define mappings  ,  : M (C) → M (C) by  ((aij )≤i,j≤ ) =  (aij )≤i,j≤ ,  =  and measures by μ({}) = μ({}) = .

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Figure 1 Refinement for two operators and a convex function f .

(I) First, we observe an example without the spectra condition (see Figure (a)). Then we obtain a refined inequality as in (), but do not have refined Jensen’s inequality. ⎛

 ⎜ If X =  ⎝ 

⎞  ⎟ ⎠ 

  

⎛

and

 ⎜ X =  ⎝ 

⎞  ⎟ ⎠ , 

  



 then X =  

  

and m = –., M = ., m = , M = , m = –., M = . (rounded to three decimal places). We have 

 (X ) +  (X )





 = 

       

      =  X +  X 

and    =         . . ¯  – δf <  X +  X + CI X = . .     .  ¯  = , <  (X ) +  (X ) + CI  . 

     X +  X

X= since C¯ = ., δf = .,



. –. –. .



.

(II) Next, we observe an example with the spectra condition (see Figure (b)). Then we obtain a series of inequalities involving refined Jensen’s inequality and its converses. ⎛ – ⎜ If X = ⎝  

 – –

⎞  ⎟ –⎠ –

⎛

and

 ⎜ X = ⎝– –

–  

⎞ – ⎟  ⎠, 

   then X =  

 



and m = –., M = –., m = ., M = ., m = –., M = ., a = ¯ = b (rounded to three decimal places). We ¯ = a, M –., b = . and we put m

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have 

  .  =  (X ) +  (X )         . – = <  X +  X – δf (a, b)X¯ – .       . – = <   X +   X – .     . –. ¯

<  (X ) +  (X ) + CI – δf (m, M)X = –. .     .  ¯ , = <  (X ) +  (X ) + CI  . 

since δf (a, b) = ., X¯ = ..

 .

  .



, δf (m, M) = ., X =



. –. –. .



and C¯ =

Applying Theorem  to f (t) = t p , we obtain the following refinement of [, Corollary .]. Corollary  Let (xt )t∈T be a field of strictly positive operators, let mx and Mx , mx ≤ Mx , be  x be defined by (). Then the bounds of the operator x = T φt (xt ) dμ(t). Let  ≤

 p φt xt dμ(t) –

T

p

 φt (xt ) dμ(t) T

  ¯ x , Mx , m, M, p)K – mp + Mp – –p (m + M)p x ≤ C(m ¯ x , Mx , m, M, p)K ≤ C(m, M, p)K ≤ C(m for p ∈/ (, ), and C(m, M, p)K ≤ c¯ (mx , Mx , m, M, p)K

  x ≤ c¯ (mx , Mx , m, M, p)K + –p (m + M)p – mp – Mp p    p ≤ φt xt dμ(t) – φt (xt ) dμ(t) ≤  T

T

for p ∈ (, ), where ⎧ p ⎪ p p ⎪ ⎨kt mx + lt – mx ¯ x , Mx , m, M, p) = C(m, M, p) C(m ⎪ ⎪ ⎩ p ktp Mx + ltp – Mx

p–

if pmx

≥ kt p ,

if

p– pmx

≤ ktp ≤ pMx ,

p–

if

p– pMx

≤ kt p ,

()

and c¯ (mx , Mx , m, M, p) equals the right-hand side in () with reverse inequality signs. C(m, M, p) is the known Kantorovich-type constant for difference (see, i.e., [, §.]): 

M p – mp C(m, M, p) = (p – ) p(M – m)

/(p–) +

Mmp – mMp M–m

for p ∈ R.

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4 Ratio-type converse inequalities In [, Theorem .] the following ratio-type converse of () is given: 

  φt f (xt ) dμ(t) ≤ max

   kf z + l f g φt (xt ) dμ(t) , g(z) T



mx ≤z≤Mx

T

()

where f is convex and g > . Applying Theorem  and Theorem , we obtain the following two refinements of (). Theorem  Let mx and Mx , mx ≤ Mx , be the bounds of the operator x = and let f : [m, M] → R, g : [mx , Mx ] → R be continuous functions. If f is convex and g > , then 

  φt f (xt ) dμ(t) ≤ max



mx ≤z≤Mx

T

 T

φt (xt ) dμ(t)

   kf z + l f g φt (xt ) dμ(t) – δf x g(z) T

()

   kf z + lf – δf m x g φt (xt ) dμ(t) , g(z) T

()

and 

  φt f (xt ) dμ(t) ≤ max



mx ≤z≤Mx

T

where x and δf are defined by () and (), respectively, and m x is the lower bound of the operator x. If f is concave, then reverse inequalities are valid in () and () with min instead of max. Proof We prove only the convex case. Let α = maxmx ≤z≤Mx { kf z +lf g(z )

kf z+lf g(z)

}. Then there is z ∈

kf z+lf g(z)

[mx , Mx ] such that α = and ≤ α for all z ∈ [mx , Mx ]. It follows that kf z + lf – α g(z ) =  and kf z + lf – α g(z) ≤  for all z ∈ [mx , Mx ]. So, max

mx ≤z≤Mx



 kf z + lf – α g(z) = .

By using (), we obtain (). Inequality () follows directly from Theorem  by putting F(u, v) = v–/ uv–/ .  Remark  () Inequality () is a refinement of () since δf x ≥ . Also, () is a refinement of () since m x ≥  and g >  implies  max

mx ≤z≤Mx

   kf z + lf – δf m x kf z + l f ≤ max . mx ≤z≤Mx g(z) g(z)

() Let the assumptions of Theorem  hold. Generally, there is no relation between the right-hand sides of inequalities () and () under the operator order (see Example ).  But, for example, if g( T φt (xt ) dμ(t)) ≤ g(z )K , where z ∈ [mx , Mx ] is the point where it k z+l achieves maxmx ≤z≤Mx { fg(z) f }, then the following order holds: 

   kf z + l f g φt f (xt ) dμ(t) ≤ max φt (xt ) dμ(t) – δf x mx ≤z≤Mx g(z) T T     kf z + lf – δf m x g φt (xt ) dμ(t) . ≤ max mx ≤z≤Mx g(z) T







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Example  Let f (t) = g(t) = t  , k ((aij )≤i,j≤ ) =  (aij )≤i,j≤ and μ({k}) = , k = , . ⎛  ⎜ If X = ⎝  

  

⎞  ⎟ ⎠ 

⎛

 ⎜ X = ⎝– –

and

–  

⎞ – ⎟  ⎠, 



. then X = 

 



and m = ., M = ., m = ., M = ., m = ., M = . (rounded to three decimal places). We have  . –. = –.      . –. < α  (X ) +  (X ) – δf x = –. .     .  , = < α  (X ) +  (X )  . 

     X +  X

since α = maxmx ≤z≤Mx {

kf z+lf g(z)

} = ., δf = ., x=

 .

()



. . .

. Further,

 . –. = –.    .  =  .   .  , =  . 

     X +  X   < α  (X ) +  (X )   < α  (X ) +  (X )

()

k z+l –δ m

x f f since α = maxmx ≤z≤Mx { f g(z) } = .. We remark that there is no relation between matrices in the right-hand sides of equalities () and ().

Remark  Similar to [, Corollary .], we can determine the constant in the RHS of (). (i) Let f be convex. We can determine the value C in 

  φt f (xt ) dμ(t) ≤ Cg

T



 φt (xt ) dμ(t)

T

as follows: • if g is convex, then ⎧ kf mx +lf –δf m x ⎪ ⎪ g(mx ) ⎪ ⎪ ⎪ ⎪ ⎨ kf z +lf –δf m x Cα =

g(z )

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ kf Mx +lf –δf m x g(Mx )

kf g(z) for every z kf z+lf –δf m x k g(z )  f g– (z ) ≤ k z +l –δ m x ≤ g+ (z ) f f f

if g– (z) ≥ if

∈ (mx , Mx ),

for some z ∈ (mx , Mx ), if g+ (z) ≤

kf g(z) kf z+lf –δf m x

()

for every z ∈ (mx , Mx );

• if g is concave, then   kf mx + lf – δf m x kf Mx + lf – δf m x C = max , . g(mx ) g(Mx )

()

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Also, we can determine the constant D in 

  φt f (xt ) dμ(t) ≤ Dg



T

 x φt (xt ) dμ(t) – δf T

in the same way as the above constant C but without m x . (ii) Let f be concave. We can determine the value c in 





  φt f (xt ) dμ(t)

φt (xt ) dμ(t) ≤

cg T

T

as follows: • if g is convex, then c is equal to the right-hand side in () with min instead of max; • if g is concave, then c is equal to the right-hand side in () with reverse inequality signs. Also, we can determine the constant d in     φt (xt ) dμ(t) – δf x ≤ φt f (xt ) dμ(t)

 dg T

T

in the same way as the above constant c but without m x . Theorem  and Remark  applied to functions f (z) = zp and g(z) = zq give the following corollary, which is a refinement of [, Corollary .]. Corollary  Let (xt )t∈T be a field of strictly positive operators, let mx and Mx , mx ≤ Mx ,  x be defined by (), m x be the lower be the bounds of the operator x = T φt (xt ) dμ(t). Let bound of the operator x and δp := mp + Mp – –p (m + M)p . (i) Let p ∈ (–∞, ] ∪ [, ∞). Then 

 p φt xt dμ(t) ≤ C 

T



q φt (xt ) dμ(t)

,

T

where the constant C  is determined as follows: • if q ∈ (–∞, ] ∪ [, ∞), then

C =

⎧ kt p mx +lt p –δp m x ⎪ ⎪ q ⎪ mx ⎨ l p –δ m

if kp

p x –q t ( q l p –δt p m x )q –q t ⎪ ⎪ ⎪ ⎩ ktp Mx +ltqp –δp m x

Mx

if if

q lt p –δp m x ≤ mx , –q kt p q lt p –δp m mx ≤ –q k p x ≤ Mx , t q lt p –δp m x ≥ Mx ; –q kp

()

t

• if q ∈ (, ), then   ktp mx + ltp – δp m x ktp q, Mx + ltp – δp m x C = max , . q q mx Mx 

Also, 

 p φt xt dμ(t) ≤ D T



q φt (xt ) dμ(t)

T

– δp x

()

Mi´ci´c et al. Journal of Inequalities and Applications 2013, 2013:353 http://www.journalofinequalitiesandapplications.com/content/2013/1/353

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holds, where D is determined in the same way as the above constant C  but without m x . (ii) Let p ∈ (, ). Then 

q

c

φt (xt ) dμ(t)



 p φt xt dμ(t),

≤

T

T

where the constant c is determined as follows: • if q ∈ (–∞, ] ∪ [, ∞), then c is equal to the right-hand side in () with min instead of max; • if q ∈ (, ), then cα is equal to the right-hand side in (). Also, 

q

d

φt (xt ) dμ(t)

 – δp x≤

T

 p φt xt dμ(t)

T

x ≥  and d is determined in the same way as the above constant holds, where δp ≤ , d but without m x . Using Theorem  and Remark  for g ≡ f and utilizing elementary calculations, we obtain the following converse of Jensen’s operator inequality.  Theorem  Let mx and Mx , mx ≤ Mx , be the bounds of the operator x = T φt (xt ) dμ(t). If f : [m, M] → R is a continuous convex function and strictly positive on [mx , Mx ], then 



  φt f (xt ) dμ(t) ≤ max

mx ≤z≤Mx

T

   kf z + lf – δf m x f φt (xt ) dμ(t) f (z) T

()

   kf z + l f f φt (xt ) dμ(t) – δf x, f (z) T

()

and 

  φt f (xt ) dμ(t) ≤ max



mx ≤z≤Mx

T

where x and δf are defined by () and (), respectively, and m x is the lower bound of the operator x. In the dual case, if f is concave, then the reverse inequalities are valid in () and () with min instead of max. Furthermore, if f is convex differentiable on [mx , Mx ], we can determine the constant  α ≡ α (m, M, mx , Mx , f ) = max

mx ≤z≤Mx

kf z + lf – δf m x f (z)



in () as follows: ⎧ k m +l –δ m x f x f f ⎪ ⎪ f (mx ) ⎪ ⎨ α =

kf z +lf –δf m x

f (z ) ⎪ ⎪ ⎪ ⎩ kf Mx +lf –δf m x f (Mx )

kf f (z) for every z ∈ (mx , Mx ), kf z+lf –δf m x k f (z )  f f  (z ) = k z +l –δ m x for some z ∈ (mx , Mx ), f f f k f (z) f  (z) ≤ k z+lf –δ m x for every z ∈ (mx , Mx ). f f f

if f  (z) ≥ if if

()

Mi´ci´c et al. Journal of Inequalities and Applications 2013, 2013:353 http://www.journalofinequalitiesandapplications.com/content/2013/1/353

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Also, if f is strictly convex twice differentiable on [mx , Mx ], then we can determine the constant  α ≡ α (m, M, mx , Mx , f ) = max

mx ≤z≤Mx

kf z + l f f (z)



in () as follows: α =

kf z + lf , f (z )

()

where z ∈ (mx , Mx ) is defined as the unique solution of the equation kf f (z) = (kf z + lf )f  (z) provided (kf mx + lf )f  (mx )/f (mx ) ≤ kf ≤ (kf Mx + lf )f  (Mx )/f (Mx ). Otherwise, z is defined as mx or Mx provided kf ≤ (kf mx + lf )f  (mx )/f (mx ) or kf ≥ (kf Mx + lf )f  (Mx )/f (Mx ), respectively. In the dual case, if f is concave differentiable, then the value α is equal to the right-hand side in () with reverse inequality signs. Also, if f is strictly concave twice differentiable, then we can determine the value α in () with z , which equals the right-hand side in () with reverse inequality signs. Remark  If f is convex and strictly negative on [mx , Mx ], then () and () are valid with min instead of max. If f is concave and strictly negative, then reverse inequalities are valid in () and (). Applying Theorem  to f (t) = t p , we obtain the following refinement of [, Corollary .]. Corollary  Let (xt )t∈T be a field of strictly positive operators, let mx and Mx , mx ≤ Mx ,  x be defined by (), m x be the lower be the bounds of the operator x = T φt (xt ) dμ(t). Let bound of the operator x and δp := mp + Mp – –p (m + M)p . If p ∈/ (, ), then  ≤

 p ¯ x , Mx , m, M, p, ) φt xt dμ(t) ≤ K(m

T

¯ x , Mx , m, M, p, ) ≤ K(m



p

≤ K(m, M, p)

p φt (xt ) dμ(t)

– δp

T

φt (xt ) dμ(t) T





p

φt (xt ) dμ(t)

()

T

and  ≤

 p ¯ x , Mx , m, M, p, m x ) φt xt dμ(t) ≤ K(m

T

¯ x , Mx , m, M, p, ) ≤ K(m



p



φt (xt ) dμ(t) T

p φt (xt ) dμ(t) T

φt (xt ) dμ(t) T

≤ K(m, M, p)



p ,

()

Mi´ci´c et al. Journal of Inequalities and Applications 2013, 2013:353 http://www.journalofinequalitiesandapplications.com/content/2013/1/353

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where

¯ x , Mx , m, M, p, c) = K(m

⎧ kt p mx +lt p –cδp ⎪ ⎪ p ⎪ mx ⎨

if

K(m, M, p, c) ⎪ ⎪ ⎪ ⎩ ktp Mx +lptp –cδp

if if

Mx

p(lt p –cδp ) mx p(lt p –cδp ) mx p(lt p –cδp ) Mx

≥ ( – p)ktp , < ( – p)ktp <

p(lt p –cδp ) , Mx

()

≤ ( – p)ktp .

K(m, M, p, c) is a generalization of the known Kantorovich constant K(m, M, p) ≡ K(m, M, p, ) (defined in [, §.]) as follows: K(m, M, p, c) :=

 p mMp – Mmp + cδp (M – m) p –  M p – mp , (p – )(M – m) p mMp – Mmp + cδp (M – m)

()

for p ∈ R and  ≤ c ≤ .. If p ∈ (, ), then 

 p ¯ x , Mx , m, M, p, ) φt xt dμ(t) ≥ k(m

T



p

T

¯ x , Mx , m, M, p, ) ≥ k(m



p φt (xt ) dμ(t)

T

 ≥ K(m, M, p)

– δp x

φt (xt ) dμ(t)

p ≥

φt (xt ) dμ(t) T

and 

 p ¯ x , Mx , m, M, p, m x ) φt xt dμ(t) ≥ k(m

T

¯ x , Mx , m, M, p, ) ≥ k(m



p φt (xt ) dμ(t)



T

T

 ≥ K(m, M, p)

p

φt (xt ) dμ(t)

φt (xt ) dμ(t)

p ≥ ,

T

¯ x , Mx , m, M, p, c) equals the right-hand side in () with reverse inequality signs. where k(m Proof The second inequalities in () and () follow directly from () and () by using () and (), respectively. The last inequality in () follows from 

 kt p z + l t p max mx ≤z≤Mx zp   kt p z + l t p = K(m, M, p). ≤ max m≤z≤M zp

¯ x , Mx , m, M, p, ) = K(m

The third inequality in () follows from ¯ x , Mx , m, M, p, m x ) = max K(m

mx ≤z≤Mx



 ktp z + ltp – δp m x ¯ x , Mx , m, M, p, ), ≤ K(m zp

since δp m x ≥  for p ∈/ (, ) and Mx ≥ mx ≥ .



Mi´ci´c et al. Journal of Inequalities and Applications 2013, 2013:353 http://www.journalofinequalitiesandapplications.com/content/2013/1/353

Figure 2 Relation between K(p, c) for p ∈ R and 0 ≤ c ≤ 0.5.

Appendix A: A new generalization of the Kantorovich constant Definition  Let h > . Further generalization of Kantorovich constant K(h, p) (given in [, Definition .]) is defined by K(h, p, c) :=

hp – h + c(hp +  – –p (h + )p )(h – ) (p – )(h – ) p  hp –  p– × p hp – h + c(hp +  – –p (h + )p )(h – )

for any real number p ∈ R and any c,  ≤ c ≤ .. The constant K(h, p, c) is sometimes denoted by K(p, c) briefly. Some of those constants are depicted in Figure . By inserting c =  in K(h, p, c), we obtain the Kantorovich constant K(h, p). The constant K(m, M, p, c) defined by () coincides with K(h, p, c) by putting h = M/m > . Lemma  Let h > . The generalized Kantorovich constant K(h, p, c) has the following properties: (i) K(h, p, c) = K( h , p, c) for all p ∈ R, (ii) K(h, , c) = K(h, , c) =  for all  ≤ c ≤ . and K(, p, c) =  for all p ∈ R, (iii) K(h, p, c) is decreasing of c for p ∈/ (, ) and increasing of c for p ∈ (, ), (iv) K(h, p, c) ≥  for all p ∈/ (, ) and  < K(h, ., ) ≤ K(h, p, c) ≤  for all p ∈ (, ), (v) K(h, p, c) ≤ hp– for all p ≥ . Proof (i) We use an easy calculation:  h–p – h– + c(h–p +  – –p (h– + )p )(h– – )  , p, c = K h (p – )(h– – ) p  h–p –  p– × p h–p – h– + c(h–p +  – –p (h– + )p )(h– – ) 

=

h – hp + c( + hp – –p (h + )p )( – h) (p – )( – h) p   – hp p– × p h – hp + c( + hp – –p (h + )p )( – h)

= K(h, p, c). (ii) Let h > . The logarithms calculation and l’Hospital’s theorem give K(h, p, b) →  as p → , K(h, p, b) →  as p →  and K(h, p, b) →  as h → +. Now using (i) we obtain (ii).

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Mi´ci´c et al. Journal of Inequalities and Applications 2013, 2013:353 http://www.journalofinequalitiesandapplications.com/content/2013/1/353

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(iii) Let h >  and  ≤ c ≤ .. dK(h, p, c) = dc



×



h+ 

p –

hp +  



hp –  p– p p p h – h + c(h +  – –p (h + )p )(h – )

p .

Since the function z → zp is convex (resp. concave) on (, ∞) if p ∈/ (, ) (resp. p ∈ (, )), p p )p ≤ h + (resp. ( h+ )p ≥ h + ) for every h > . Then dK(h,p,c) ≤  if p ∈/ (, ) and then ( h+   dc dK(h,p,c) ≥  if p ∈ (, ), which gives that K(h, p, c) is decreasing of c if p ∈/ (, ) and indc creasing of c if p ∈ (, ). (iv) Let h >  and  ≤ c ≤ .. If p >  then < ≤

(p – )(h – ) hp – h + c(hp +  – –p (h + )p )(h – ) p– hp –  p hp – h + c(hp +  – –p (h + )p )(h – )

implies

hp

(p – )(h – ) +  – –p (h + )p )(h – )

– h + c(hp 

≤

hp –  p– p hp – h + c(hp +  – –p (h + )p )(h – )

p ,

which gives K(h, p, c) ≥ . Similarly, K(h, p, c) ≥  if p <  and K(h, p, c) ≤  if p ∈ (, ). Next, using (iii) and [, Theorem .(iv)], K(h, p, c) ≥ K(h, p, ) ≥ K(h, ., ) for p ∈ (, ).  (v) Let p ≥ . Using (iii) and [, Theorem .(vi)], K(h, p, c) ≤ K(h, p, ) ≤ hp– .

Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Author details 1 Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Luˇci´ca 5, Zagreb, 10000, Croatia. 2 Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovi´ca 30, Zagreb, 10000, Croatia. 3 Faculty of Science, Department of Mathematics, University of Split, Teslina 12, Split, 21000, Croatia. Received: 27 November 2012 Accepted: 9 July 2013 Published: 29 July 2013 References 1. Davis, C: A Schwarz inequality for convex operator functions. Proc. Am. Math. Soc. 8, 42-44 (1957) 2. Choi, MD: A Schwarz inequality for positive linear maps on C ∗ -algebras. Ill. J. Math. 18, 565-574 (1974) 3. Hansen, F, Pedersen, GK: Jensen’s inequality for operators and Löwner’s theorem. Math. Ann. 258, 229-241 (1982) 4. Hansen, F, Pedersen, GK: Jensen’s operator inequality. Bull. Lond. Math. Soc. 35, 553-564 (2003) 5. Mond, B, Peˇcari´c, J: On Jensen’s inequality for operator convex functions. Houst. J. Math. 21, 739-754 (1995) 6. Furuta, T, Mi´ci´c Hot, J, Peˇcari´c, J, Seo, Y: Mond-Peˇcari´c Method in Operator Inequalities. Monographs in Inequalities, vol. 1. Element, Zagreb (2005) 7. Hansen, F, Peˇcari´c, J, Peri´c, I: Jensen’s operator inequality and its converses. Math. Scand. 100, 61-73 (2007) 8. Abramovich, S, Jameson, G, Sinnamon, G: Refining Jensen’s inequality. Bull. Math. Soc. Sci. Math. Roum. 47, 3-14 (2004) 9. Dragomir, SS: A new refinement of Jensen’s inequality in linear spaces with applications. Math. Comput. Model. 52, 1497-1505 (2010)

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doi:10.1186/1029-242X-2013-353 Cite this article as: Mi´ci´c et al.: Refined converses of Jensen’s inequality for operators. Journal of Inequalities and Applications 2013 2013:353.

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