Recent Research on Jensen's Inequality for Oparators

Let f be an operator convex function defined on an interval I. Ch. Davis [1] proved1 a Schwarz inequality f (φ(x)) ≤ φ ( f (x)) (2) 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 M. D. Choi [2] noted that it is enough to assume that φ is unital and positive. In fact, the restriction of φ to the commutative C∗-algebra generated by x is automatically completely positive by a theorem of Stinespring.


Introduction
The self-adjoint operators on Hilbert spaces with their numerous applications play an important part in the operator theory.The bounds research for self-adjoint operators is a very useful area of this theory.There is no better inequality in bounds examination than Jensen's inequality.It is an extensively used inequality in various fields of mathematics.
Let I be a real interval of any type.A continuous function f : I → R is said to be operator convex if holds for each λ ∈ [0, 1] and every pair of self-adjoint operators x and y (acting) on an infinite dimensional Hilbert space H with spectra in I (the ordering is defined by setting x ≤ y if y − x is positive semi-definite).
Let f be an operator convex function defined on an interval I. Ch.Davis [1] proved 1 a Schwarz 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 M. D. Choi [2] noted that it is enough to assume that φ is unital and positive.
In fact, the restriction of φ to the commutative C * -algebra generated by x is automatically completely positive by a theorem of Stinespring.
F. Hansen and G. K. Pedersen [3] proved a Jensen type inequality for operator convex functions f defined on an interval I = [0, α) (with α ≤ ∞ and f (0) ≤ 0) and self-adjoint operators x 1 , . . ., x n with spectra in I assuming that ∑ n i=1 a * i a i = 1.The restriction on the interval and the requirement f (0) ≤ 0 was subsequently removed by B. Mond and J. Pečarić in [4], cf. also [5].
The inequality (3) is in fact just a reformulation of (2) although this was not noticed at the time.It is nevertheless important to note that the proof given in [3] and thus the statement of the theorem, when restricted to n × n matrices, holds for the much richer class of 2n × 2n matrix convex functions.Hansen and Pedersen used (3) to obtain elementary operations on functions, which leave invariant the class of operator monotone functions.These results then served as the basis for a new proof of Löwner's theorem applying convexity theory and Krein-Milman's theorem.
B. Mond and J. Pečarić [6] proved the inequality for operator convex functions f defined on an interval I, where φ i : B(H) → B(K) are unital positive linear mappings, x 1 , . . ., x n are self-adjoint operators with spectra in I and w 1 , . . ., w n are are non-negative real numbers with sum one.
Also, B. Mond, J. Pečarić, T. Furuta et al. [6][7][8][9][10][11] observed conversed of some special case of Jensen's inequality.So in [10] presented the following generalized converse of a Schwarz inequality ( 2) for convex functions 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, matrix non-decreasing in u, U ⊃ f [m, M], V ⊃ g[m, M], φ : H n → H ñ is a unital positive linear mapping and A is a Hermitian matrix with spectrum contained in [m, M].
There are a lot of new research on the classical Jensen inequality (4) and its reverse inequalities.For example, J.I. Fujii et all. in [12,13] expressed these inequalities by externally dividing points.

Classic results
In this section we present a form of Jensen's inequality which contains (2), ( 3) and (4) as special cases.Since the inequality in (4) was the motivating step for obtaining converses of Jensen's inequality using the so-called Mond-Pečarić method, we also give some results pertaining to converse inequalities in the new formulation.
We recall some definitions.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 (x t ) t∈T of operators in A is continuous if the function t → x t is norm continuous on T. If in addition μ is a Radon measure on T and the function t → x t is integrable, then we can form the Bochner integral T x t dμ(t), which is the unique element in A such that for every linear functional ϕ in the norm dual A * .
Assume furthermore that there is a field (φ t ) t∈T of positive linear mappings φ t : A → B from A to another C * -algebra B of operators on a Hilbert space K.We recall that a linear mapping φ t : A → B is said to be a positive mapping if φ t (x t ) ≥ 0 for all x t ≥ 0. We say that such a field is continuous if the function t → φ t (x) is continuous for every x ∈ A. Let the C * -algebras include the identity operators and the function t → φ t (1 H ) be integrable with Let B(H) be the C * -algebra of all bounded linear operators on a Hilbert space H.We define bounds of an operator x ∈ B(H) by For an operator x ∈ B(H) we define operators |x|, Obviously, if x is self-adjoint, then |x| = (x 2 ) 1/2 and x + , x − ≥ 0 (called positive and negative parts of x = x + − x − ).

Jensen's inequality with operator convexity
Firstly, we give a general formulation of Jensen's operator inequality for a unital field of positive linear mappings (see [14]).
Theorem 1.Let f : I → R be an operator convex function defined on an interval I and let A and B be unital C * -algebras acting on a Hilbert space H and K respectively.If (φ t ) t∈T is a unital field of positive linear mappings φ t : A → B defined on a locally compact Hausdorff space T with a bounded Radon measure μ, then the inequality holds for every bounded continuous field (x t ) t∈T of self-adjoint elements in A with spectra contained in I.
Proof.We first note that the function t → φ t (x t ) ∈ B is continuous and bounded, hence integrable with respect to the bounded Radon measure μ.Furthermore, the integral is an element in the multiplier algebra M(B) acting on K.We may organize the set CB(T, A) of bounded continuous functions on T with values in A as a normed involutive algebra by applying the point-wise operations and setting and it is not difficult to verify that the norm is already complete and satisfy the C * -identity.
In fact, this is a standard construction in C * -algebra theory.It follows that f ((x t ) t∈T ) = ( f (x t )) t∈T .We then consider the mapping defined by setting and note that it is a unital positive linear map.Setting x = (x t ) t∈T ∈ CB(T, A), we use inequality (2) to obtain but this is just the statement of the theorem.

Converses of Jensen's inequality
In the present context we may obtain results of the Li-Mathias type cf.[15, Chapter 3] and [16,17].
holds for every operator convex function h Applying RHS of (8) for a convex function f (or LHS of (8) for a concave function f ) we obtain the following generalization of (5).
Theorem 3. Let (x t ) t∈T , m x , M x and (φ t ) t∈T be as in Theorem 2. Let f In the dual case (when f is concave) the opposite inequalities hold in (10) with inf instead of sup.
Proof.We prove only the convex case.For convex f the inequality f (z 9) we obtain (10).
Numerous applications of the previous theorem can be given (see [15]).Applying Theorem 3 for the function F(u, v) = u − αv and k = 1, we obtain the following generalization of [15,Theorem 2.4].Corollary 4. Let (x t ) t∈T , m x , M x be as in Theorem 2 and (φ t ) t∈T be a unital field of positive linear mappings where If furthermore αg is strictly convex differentiable, then the constant C ≡ C(m, M, f , g, α) can be written more precisely as where In the dual case (when f is concave and αg is strictly concave differentiable) the opposite inequalities hold in (11) with min instead of max with the opposite condition while determining z 0 .

Inequalities with conditions on spectra
In this section we present Jensens's operator inequality for real valued continuous convex functions with conditions on the spectra of the operators.A discrete version of this result is given in [18].Also, we obtain generalized converses of Jensen's inequality under the same conditions.
Operator convexity plays an essential role in (2).In fact, the inequality (2) will be false if we replace an operator convex function by a general convex function.For example, M.D.
Choi in [2, Remark 2.6] considered the function f (t) = t 4 which is convex but not operator convex.He demonstrated that it is sufficient to put dimH = 3, so we have the matrix case as follows.Let Φ : ) and no relation between Φ(A) 4 and Φ(A 4 ) under the operator order.
Example 5.It appears that the inequality (7) will be false if we replace the operator convex function by a general convex function.We give a small example for the matrix cases and T = {1, 2}.We define mappings Given the above, there is no relation between (Φ 1 (X 1 ) + Φ 2 (X 2 )) 4 and under the operator order.We observe that in the above case the following stands X = Φ 1 (X 1 ) + So we have that an inequality of type (7) now is valid.In the above case the following stands

Jensen's inequality without operator convexity
It is no coincidence that the inequality ( 7) is valid in Example 18-II).In the following theorem we prove a general result when Jensen's operator inequality (7) holds for convex functions.
Theorem 6.Let (x t ) 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 m t and M t , m t ≤ M t , be the bounds of x t , t ∈ T. Let (φ t ) t∈T be a unital field of positive linear mappings holds for every continuous convex function f : I → R provided that the interval I contains all m t , M t .
If f : I → R is concave, then the reverse inequality is valid in (12).
Proof.We prove only the case when f is a convex function.If we denote m = inf , then by using functional calculus, it follows from ( 13) On the other hand, since m t 1 H ≤ x t ≤ M t 1 H , t ∈ T, then by using functional calculus, it follows from ( 14) Applying a positive linear mapping φ t and summing, we obtain Combining the two inequalities ( 15) and ( 16), we have the desired inequality (12).
where l is the subdifferential of f .Since m1 H ≤ x t ≤ M1 H , t ∈ T, then by using functional calculus, applying a positive linear mapping φ t and summing, we obtain from ( 17) which is the desired inequality (12).
Putting φ t (y) = a t y for every y ∈ A, where a t ≥ 0 is a real number, we obtain the following obvious corollary of Theorem 6.

Converses of Jensen's inequality with conditions on spectra
Using the condition on spectra we obtain the following extension of Theorem 3.

is bounded and operator monotone in the first variable, then
inf In the dual case (when f is concave) the opposite inequalities hold in (19) by replacing inf and sup with sup and inf, respectively.
Proof.We prove only LHS of (19).It follows from ( 14) (compare it to ( 16)) In the dual case (when f is concave) the opposite inequalities hold in (20) by replacing min and max with max and min, respectively.If additionally g > 0 on [m x , M x ], then the opposite inequalities also hold in (21) by replacing min and max with max and min, respectively.

Refined Jensen's inequality
In this section we present a refinement of Jensen's inequality for real valued continuous convex functions given in Theorem 6.A discrete version of this result is given in [19].
To obtain our result we need the following two lemmas.
Lemma 10.Let f be a convex function on an interval I, m, M ∈ I and p 1 , Proof.These results follows from [20, Theorem 1, p. 717].
Lemma 11.Let x be a bounded self-adjoint elements in a unital C * -algebra A of operators on some Hilbert space H.If the spectrum of x is in [m, M], for some scalars m < M, then holds for every continuous convex (resp.concave) function f : [m, M] → R, where Proof.We prove only the convex case.It follows from ( 22) that for every Then by using (24) for Finally we use the continuous functional calculus for a self-adjoint operator x: f , g ∈ C(I), Sp(x) ⊆ I and f ≤ g on I implies f (x) ≤ g(x); and h(z) = |z| implies h(x) = |x|.
Then by using (25) we obtain the desired inequality (23).
holds, where Proof.We prove only the convex case.Since where δ f and x are defined by (28).
, t ∈ T Applying a positive linear mapping φ t , integrating and adding −δ f x, we obtain Combining the two inequalities ( 29) and (30), we have LHS of (26).Since δ f ≥ 0 and x ≥ 0, then we have RHS of (26).
If m < M and m x = M x , then the inequality (26) holds, but δ f (m x , M x ) x(m x , M x ) is not defined (see Example 13 I) and II)).

Example 13.
We give examples for the matrix cases and T = {1, 2}.Then we have refined inequalities given in Fig. 2. We put f (t) = t 4 which is convex but not operator convex in (26).Also, we define mappings

I) First, we observe an example when δ f X is equal to the difference RHS and LHS of Jensen's inequality. If X
We also put m = −3 and M = 2.We obtain We remark that in this case m x = M x = −1/2 and X(m x , M x ) is not defined.

II) Next, we observe an example when δ f X is not equal to the difference RHS and LHS of Jensen's inequality and m
In this case x(m x , M x ) is not defined, since m x = M x = 1/2.We have and putting m = −1, M = 2 we obtain δ f = 135/8, X = I 2 /2 which give the following improvement 1 0 0 1 III) Next, we observe an example with matrices that are not special.If 3446 and we put m = m, M = M (rounded to four decimal places).We have 201 Recent Research on Jensen's Inequality for Operators and its improvement 1.5787 0 0 0.6441 (rounded to four decimal places), since δ f = 3.1574, X = 0.5 0 0 0.2040 .But, if we put m = m x = 0, M = M x = 0.5, then X = 0, so we do not have an improvement of Jensen's inequality.Also, if we put holds, where δ f is defined by (28)

Extension Jensen's inequality
In this section we present an extension of Jensen's operator inequality for n−tuples of self-adjoint operators, unital n−tuples of positive linear mappings and real valued continuous convex functions with conditions on the spectra of the operators.In a discrete version of Theorem 6 we prove that Jensen's operator inequality holds for every continuous convex function and for every n−tuple of self-adjoint operators (A 1 , . . ., A n ), for every n−tuple of positive linear mappings (Φ 1 , . . ., Φ n ) in the case when the interval with bounds of the operator A = ∑ n i=1 Φ i (A i ) has no intersection points with the interval with bounds of the operator A i for each i = 1, . . ., n, i.e. when (m A , M A ) ∩ [m i , M i ] = ∅ for i = 1, . . ., n, where m A and M A , m A ≤ M A , are the bounds of A, and m i and M i , m i ≤ M i , are the bounds of A i , i = 1, . . ., n.It is interesting to consider the case when (m A , M A ) ∩ [m i , M i ] = ∅ is valid for several i ∈ {1, . . ., n}, but not for all i = 1, . . ., n.We study it in the following theorem (see [21]).Theorem 15.Let (A 1 , . . ., A n ) be an n−tuple of self-adjoint operators A i ∈ B(H) with the bounds m i and M i , m i ≤ M i , i = 1, . . ., n.Let (Φ 1 , . . ., Φ n ) be an n−tuple of positive linear mappings Φ i : and one of two equalities holds for every continuous convex function f : I → R provided that the interval I contains all m i , M i , i = 1, . . ., n.If f : I → R is concave, then the reverse inequality is valid in (31).
Proof.We prove only the case when f is a convex function.Let us denote Since Applying a positive linear mapping Φ i and summing, we obtain Similarly to (34) in the case Combining ( 34) and ( 35) and taking into account that A = B, we obtain It follows which gives the desired double inequality (31 On the other hand, since f is convex on I, we have where l is the subdifferential of f .Replacing z by A i for i = n 1 + 1, . . ., n, applying Φ i and summing, we obtain from (38) and (37) So (36) holds again.The remaining part of the proof is the same as in the case a).
Remark 16.We obtain the equivalent inequality to the one in Theorem 15 in the case when ∑ n i=1 Φ i (1 H ) = γ 1 K , for some positive scalar γ.If α + β = γ and one of two equalities holds for every continuous convex function f .Remark 17.Let the assumptions of Theorem 15 be valid.
1. We observe that the following inequality holds for every continuous convex function f : Indeed, by the assumptions of Theorem 15 we have So we can apply Theorem 6 on operators A n 1 +1 , . . ., A n and mappings 1  β Φ i and obtain the desired inequality.

2.
We denote by m C and M C the bounds of . ., n 1 or f is an operator convex function on [m, M], then the double inequality (31) can be extended from the left side if we use Jensen's operator inequality (see [16,Theorem 2.1]) for every α ∈ (0, 1).We observe that f (t) = t 4 is not operator convex and (m and one of two equalities holds for every continuous convex function f : I → R provided that the interval I contains all m i , M i , i = 1, . . ., n. If f : I → R is concave, then the reverse inequality is valid in (39).
As a special case of Corollary where m A and M A , m A ≤ M A , are the bounds of A = ∑ n i=1 α i A i , then holds for every continuous convex function f : I → R provided that the interval I contains all m i , M i .
In this section we present an extension of the refined Jensen's inequality obtained in Section 4 and a refinement of the same inequality obtained in Section 5.
Theorem 21.Let (A 1 , . . ., A n ) be an n−tuple of self-adjoint operators A i ∈ B(H) with the bounds m i and M i , m i ≤ M i , i = 1, . . ., n.Let (Φ 1 , . . ., Φ n ) be an n−tuple of positive linear mappings

, n, and m < M and one of two equalities
holds for every continuous convex function f : I → R provided that the interval I contains all m i , M i , i = 1, . . ., n, where Proof.We prove only the convex case.Let us denote H . Applying a positive linear mapping Φ i and summing, we obtain Combining ( 46) and (47) and taking into account that A = B, we obtain which gives the following double inequality Adding βδ f A in the above inequalities, we get Now, we remark that δ f ≥ 0 and A ≥ 0. (Indeed, since f is convex, then f (( m + M)/2) ≤ ( f ( m) + f ( M))/2, which implies that δ f ≥ 0. Also, since Consequently, the following inequalities hold, which with (49) proves the desired series inequalities (44).
Example 22.We observe the matrix case of Theorem 21 for f (t) = t 4 , which is the convex function but not operator convex, n = 4, n 1 = 2 and the bounds of matrices as in Fig. 3.We show an example

Figure 2 .
Figure 2. Refinement for two operators and a convex function f

Figure 3 . 1 αI 3 |⎠
Figure 3.An example a convex function and the bounds of four operators

operator x = T φ t (x t ) dμ(t) and f :
Theorem 2. Let T be a locally compact Hausdorff space equipped with a bounded Radon measure μ.Let (x t ) t∈T be a bounded continuous field of self-adjoint elements in a unital C * -algebra A with spectra in [m, M], m < M. Furthermore, let (φ t ) t∈T be a field of positive linear mappings φ t

Corollary 7 .
Let (x t ) 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 m t and M t , m t ≤ M t , be the bounds of x t , t ∈ T. Let (a t ) t∈T be a continuous field of nonnegative real numbers such that T a t dμ(t) = 1.If (m x , M x ) ∩ [m t , M t ] = ∅,holds for every continuous convex function f : I → R provided that the interval I contains all m t , M t .