Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 26, pp. 192-200, May 2013
ELA
http://math.technion.ac.il/iic/ela
BOUNDS FOR AN OPERATOR CONCAVE FUNCTION∗
SEVER S. DRAGOMIR† , JUN ICHI FUJII‡ , AND YUKI SEO§
Abstract. Let Φ and Ψ be unital positive linear maps satisfying some conditions with respect
to positive scalars α and β. It is shown that if a real valued function f is operator concave on an
interval J, then
β (f (Ψ(A)) − Ψ(f (A))) ≤ f (Φ(A)) − Φ(f (A)) ≤ α (f (Ψ(A)) − Ψ(f (A)))
for every selfadjoint operator A with spectrum σ(A) ⊂ J. Moreover, an external version of estimates
above is presented.
Key words. Operator concave function, Davis-Choi-Jensen inequality, Positive linear map.
AMS subject classifications. 47A63.
1. Introduction. Let Φ be a unital positive linear map from B(H) to B(K),
where B(H) is the C∗ -algebra of all bounded linear operators on a Hilbert space
H. The Davis-Choi-Jensen inequality [1, 2] states that if a real-valued function f is
operator concave on an interval J, then
(1.1)
Φ(f (A)) ≤ f (Φ(A))
for every selfadjoint operator A with spectrum σ(A) ⊂ J. Though in the case of
concave function the inequality (1.1) does not hold in general, we have the following
estimate [7]: If f is concave and A is a selfadjoint operator on H such that mI ≤ A ≤
M I for some scalars m < M , then
(1.2)
− µ(m, M, f )I ≤ f (Φ(A)) − Φ(f (A)) ≤ µ(m, M, f )I
for all unital positive linear maps Φ where the bound µ(m, M, f ) of f is defined by
f (M ) − f (m)
(1.3) µ(m, M, f ) = max f (t) −
(t − m) − f (m) : t ∈ [m, M ] .
M −m
We note that bounds of (1.2) are sharp.
∗ Received by the editors on September 14, 2012. Accepted for publication on March 22, 2013.
Handling Editor: Moshe Goldberg.
† Mathematics, School of Engineering and Science, Victoria University, PO Box 14428, Melbourne
City, MC 8001, Australia ([email protected]). School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits, 2050, South Africa
([email protected]).
‡ Department of Art and Sciences (Information Science), Osaka Kyoiku University, Asahigaoka,
Kashiwara, Osaka 582-8582, Japan ([email protected]).
§ Department of Mathematics Education, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka
582-8582, Japan ([email protected]).
192
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 26, pp. 192-200, May 2013
ELA
http://math.technion.ac.il/iic/ela
193
Bounds for an Operator Concave Function
In [3], the first author showed the following estimate for the normalized Jensen
functional: If a real-valued function f is concave on a convex set C, then for each
Pn
Pn
positive n-tuples (p1 , . . . , pn ) and (q1 , . . . , qn ) with i=1 pi = 1 and i=1 qi = 1
!
!
!
!
n
n
n
n
X
X
X
X
pi
pi f (xi )
pi xi −
qi f (xi ) ≤ f
min
qi xi −
f
1≤i≤n
qi
i=1
i=1
i=1
i=1
!
!
n
n
X
X
pi
≤ max
qi f (xi )
qi xi −
f
1≤i≤n
qi
i=1
i=1
for all (x1 , . . . , xn ) ∈ C n .
In this note, based on the idea of [3], we shall provide new bounds for the difference
of the Davis-Choi-Jensen inequality. Among others, we show that if Φ is a unital
positive linear map and f is operator concave on an interval [m, M ], then
m+M
f (m) + f (M )
f (Φ(A)) − Φ(f (A)) ≤ 2 f
−
I
2
2
for every selfadjoint operator A such that mI ≤ A ≤ M I for some scalars m < M .
Moreover, we discuss an external version of the Davis-Choi-Jensen inequality.
2. Davis-Choi-Jensen inequality. Let Φ and Ψ be two positive linear maps
from B(H) to B(K). Φ is said to be α-upper dominant by Ψ if there exists α > 0 such
that αΨ ≥ Φ. Similarly Φ is said to be β-lower dominant by Ψ if there exists β > 0
such that Φ ≥ βΨ. Moreover, Φ is (α, β)-dominant by Ψ if Φ is α-upper and β-lower
dominant by Ψ. The vector (p1 , . . . , pn ) is said to be a weight vector if pi > 0 for all
Pn
i = 1, . . . , n and i=1 pi = 1. For example, we put two positive linear maps Φ and
Ψ : B(H) ⊕ · · · ⊕ B(H) 7→ B(H) as follows:
Φ(A1 ⊕ · · · ⊕ An ) =
n
X
pi Ai
and Ψ(A1 ⊕ · · · ⊕ An ) =
n
X
qi Ai ,
i=1
i=1
where (p1 , . . . , pn ) and (q1 , . . . , qn ) are weight vectors. If we put α = max1≤i≤n { pqii }
and β = min1≤i≤n { pqii }, then it follows that Φ is (α, β)-dominant by Ψ. In fact, we
have
n
X
pi
(α − )qi Ai
(αΨ − Φ)(A1 ⊕ · · · ⊕ An ) =
qi
i=1
and
(Φ − βΨ)(A1 ⊕ · · · ⊕ An ) =
n
X
pi
( − β)qi Ai .
qi
i=1
Therefore, αΨ − Φ and Φ − βΨ are positive linear maps.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 26, pp. 192-200, May 2013
ELA
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Sever S. Dragomir, Jun Ichi Fujii, and Yuki Seo
Firstly, we provide the following estimates associated with the Davis-Choi-Jensen
inequality for two unital positive linear maps:
Theorem 2.1. Let Φ and Ψ be two unital positive linear maps from B(H) to
B(K) such that Φ is (α, β)-dominant by Ψ. If f is operator concave on an interval
J, then
(2.1) β (f (Ψ(A)) − Ψ(f (A))) ≤ f (Φ(A)) − Φ(f (A)) ≤ α (f (Ψ(A)) − Ψ(f (A)))
for every selfadjoint operator A with spectrum σ(A) ⊂ J.
Proof. If we put Φ0 (X) = α1 X, then Φ0 is a positive linear map. Since Φ is
α-upper dominant by Ψ, we have Ψ − α1 Φ is positive and (Ψ − α1 Φ)(I) + Φ0 (I) = I.
Therefore, by (1.1), we have
1
1
1
f (Ψ(A)) = f (Ψ(A) − Φ(A) + Φ(A)) = f (Ψ − Φ)(A) + Φ0 (Φ(A))
α
α
α
1
≥ (Ψ − Φ)(f (A)) + Φ0 (f (Φ(A)))
α
1
1
= Ψ(f (A)) − Φ(f (A)) + f (Φ(A))
α
α
and this implies the second inequality of (2.1).
Similarly, if we put Φ1 (X) = βX, then it follows that
f (Φ(A)) = f (Φ(A) − βΨ(A) + βΨ(A)) = f ((Φ − βΨ)(A) + Φ1 (Ψ(A)))
≥ (Φ − βΨ)(f (A)) + Φ1 (f (Ψ(A))) = Φ(f (A)) − βΨ(f (A)) + βf (Ψ(A)).
By Theorem 2.1, we have the following corollary as an operator concave version
of [3, Theorem 1], see also [4].
Corollary 2.2. Let (p1 , . . . , pn ) and (q1 , . . . , qn ) be weight vectors. If f is
operator concave on an interval J, then
!
!
n
n
X
X
qi f (Ai )
qi Ai −
β f
i=1
i=1
≤f
n
X
pi Ai
i=1
!
−
n
X
pi f (Ai ) ≤ α f
n
X
i=1
i=1
qi Ai
!
−
n
X
i=1
!
qi f (Ai )
for all selfadjoint operators A1 , . . . , An such that σ(Ai ) ⊂ J for all i = 1, . . . , n, where
α = max1≤i≤n { pqii } and β = min1≤i≤n { pqii }.
In particular,
n min {pi } f
1≤i≤n
n
X
1
Ai
n
i=1
!
!
n
X
1
−
f (Ai )
n
i=1
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
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Bounds for an Operator Concave Function
≤f
n
X
pi Ai
i=1
!
−
n
X
pi f (Ai ) ≤ n max {pi } f
1≤i≤n
i=1
n
X
1
Ai
n
i=1
!
−
!
n
X
1
f (Ai ) .
n
i=1
The following corollary is a two variable version of Theorem 2.1.
Corollary 2.3. Let Φ, Ψ, Φ′ and Ψ′ be positive linear maps from B(H) to
B(K) such that Φ(I) + Ψ(I) = I and Φ′ (I) + Ψ′ (I) = I and Φ is (α, β)-dominant by
Φ′ and Ψ is (α, β)-dominant by Ψ′ . If a real-valued function f is operator concave on
an interval J, then
β (f (Φ′ (A) + Ψ′ (B)) − (Φ′ (f (A)) + Ψ′ (f (B)))
≤ f (Φ(A) + Ψ(B)) − (Φ(f (A)) + Ψ(f (B)))
≤ α (f (Φ′ (A) + Ψ′ (B)) − (Φ′ (f (A)) + Ψ′ (f (B)))
for all selfadjoint operators A and B with σ(A), σ(B), σ(Φ(A)+Ψ(B)) and σ(Φ′ (A)+
Ψ′ (B)) ⊂ J.
Remark 2.4. Similarly, we have the following n-variable version of Corollary 3.
Pn
Let {Φi } and {Φ′i } be positive linear maps from B(H) to B(K) such that i=1 Φi (I) =
Pn
′
′
i=1 Φi (I) = I and Φi is (α, β)-dominant by Φi for i = 1, . . . , n. If a real-valued
function f is operator concave on an interval J, then
!
!
!
n
n
n
n
X
X
X
X
′
′
β f
Φi (Ai ) −
Φi (f (Ai )) ≤ f
Φi (Ai ) −
Φi (f (Ai ))
i=1
≤α f
n
X
i=1
i=1
!
Φ′i (Ai )
−
n
X
i=1
i=1
i=1
!
Φ′i (f (Ai ))
Pn
for all selfadjoint operators A and B with σ(A), σ(B), σ( i=1 Φi (A)) and
Pn
σ( i=1 Φ′i (A)) ⊂ J.
In the case of a concave function, we have no relation between Φ(f (A)) and
f (Φ(A)). Though we have the estimate of (1.2), we provide new bounds for the
difference of the Davis-Choi-Jensen inequality by means of the difference of concavity.
Theorem 2.5. Let Φ be a unital positive linear map from B(H) to B(K). If a
real-valued function f (t) is concave on [m, M ], then
2
f (m) + f (M )
m+M
−
f
−
Φ(F (A))
M −m
2
2
f (m) + f (M )
m+M
2
f
−
F (Φ(A))
≤ f (Φ(A)) − Φ(f (A)) ≤
M −m
2
2
for every selfadjoint operator A such that mI ≤ A ≤ M I for some scalars m < M ,
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 26, pp. 192-200, May 2013
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196
Sever S. Dragomir, Jun Ichi Fujii, and Yuki Seo
where a real-valued function F (t) on [m, M ] is defined by
M − m M + m F (t) =
.
+ t −
2
2 Proof. Since Φ is a unital positive linear map and f is concave on [m, M ], we have
f (M ) − f (m)
M f (m) − mf (M )
Φ(f (A)) ≥ Φ
A+
I
M −m
M −m
f (M ) − f (m)
M f (m) − mf (M )
=
Φ(A) +
I.
M −m
M −m
On the other hand, it follows from (1.4) that
M f (m) − mf (M )
f (M ) − f (m)
t+
f (t) −
M −m
M −m
M −t
t−m
M −t
t−m
=f
m+
M −
f (m) +
f (M )
M −m
M −m
M −m
M −m
f (m) + f (M )
m+M
2
−
max {M − t, t − m} f
≤
M −m
2
2
2
f (m) + f (M )
m+M
=
F (t),
−f
M −m
2
2
and this implies
2
f (Φ(A)) − Φ(f (A)) ≤
M −m
f (m) + f (M )
−f
2
m+M
2
F (Φ(A)).
For the first half of Theorem 2.5, we have
M f (m) − mf (M )
f (M ) − f (m)
Φ(A) +
I − Φ(f (A))
M −m
M −m
f (M ) − f (m)
M f (m) − mf (M )
= Φ(
A+
I − f (A))
M −m
M −m
f (m) + f (M )
m+M
2
Φ(F (A)).
−f
≥−
M −m
2
2
f (Φ(A)) − Φ(f (A)) ≥
1
F (Φ(A)) ≤ I in Theorem 2.5, we have an evaluation type by the
Since M−m
middle point:
Corollary 2.6. Let Φ, f and A be as in Theorem 2.5. Then
f (m) + f (M )
m+M
−
I.
f (Φ(A)) − Φ(f (A)) ≤ 2 f
2
2
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Bounds for an Operator Concave Function
The following corollary is another expression of (1.2).
Corollary 2.7. Let Φ, f and A be as in Theorem 2.5. Then
f (m) + f (M )
f (m) + f (M )
˜
˜
− fmax −
I ≤ f (Φ(A)) − Φ(f (A)) ≤ fmax −
I,
2
2
(m)
(M)−f (m))
where f˜(t) = f (t) − f (M)−f
t + (M+m)(f
and f˜max = max{f˜(t) : m ≤ t ≤
M−m
2(M−m)
M }.
Proof. Since
f (t) −
f (M ) − f (m)
M f (m) − mf (M )
f (m) + f (M )
t−
= f˜(t) −
M −m
M −m
2
f
(m)
+ f (M )
≤ f˜max −
,
2
it follows from the concavity of f that
M f (m) − mf (M )
f (M ) − f (m)
Φ(A) −
I
f (Φ(A)) − Φ(f (A)) ≤ f (Φ(A)) −
M −m
M −m
f (m) + f (M )
I.
≤ f˜max −
2
On the other hand, by the Stinespring decomposition theorem [10], Φ restricted to a
C∗ -algebra C ∗ (A) generated by A and I admits a decomposition Φ(X) = C ∗ φ(X)C
for all X ∈ C ∗ (A), where φ is a ∗-representation of C ∗ (A) ⊂ B(H) and C is a bounded
linear operator from K to a Hilbert space K ′ . Since Φ is unital, we have C ∗ C = I.
For every unit vector x ∈ K,
hf (Φ(A))x, xi − hΦ(f (A))x, xi = hf (C ∗ φ(A)C)x, xi − hC ∗ φ(f (A))Cx, xi
= hf (φ(A))Cx, Cxi − hf (C ∗ φ(A)C)x, xi
≤ f (hφ(A)Cx, Cxi) − hf (C ∗ φ(A)C)x, xi
f (M ) − f (m) ∗
M f (m) − mf (M )
≤ f (hC ∗ φ(A)Cx, xi) −
hC φ(A)Cx, xi −
M −m
M −m
f
(m)
+
f
(M
)
.
≤ f˜max −
2
Therefore, we have
Φ(f (A)) − f (Φ(A)) ≤
f (m) + f (M )
f˜max −
2
and this implies the first half part of the desired inequality.
I
Electronic Journal of Linear Algebra ISSN 1081-3810
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Sever S. Dragomir, Jun Ichi Fujii, and Yuki Seo
3. External version of Davis-Choi-Jensen inequality. In this section, we
consider bounds of operator concavity in terms of an external formula. A real-valued
continuous function f on J is operator concave if and only if
(3.1)
f ((1 + p)A − pB) ≤ (1 + p)f (A) − pf (B)
for all p > 0 and all selfadjoint operators A and B with σ(A), σ(B) and σ((1 +
p)A − pB) ⊂ J. Then we have the following external version of the Jensen operator
inequality: If f is operator concave, then
!
n
n
n
n
X
X
X
X
(3.2)
pi f (Bi )
pi )f (A) −
pi Bi ≤ (1 +
pi )A −
f (1 +
i=1
i=1
i=1
i=1
for all selfadjoint operators A and Bi (i = 1, . . . , n) with σ(A), σ(Bi ) and σ((1 +
Pn
Pn
i=1 pi )A −
i=1 pi Bi ) ⊂ J, also see [5, 9].
For a real-valued continuous function f , we define the following notation
1
1
1
1
A σf B = A 2 f (A− 2 BA− 2 )A 2
for positive invertible A and selfadjoint B, also see [8].
Let Φ be a positive linear map from B(H) to B(K). In [6], we show the following external version of the Davis-Choi-Jensen inequality: Let f be a real-valued
continuous function on an interval J. Then f is operator convcave if and only if
(3.3)
f (Φ(A) − Ψ(B)) ≤ Φ(I) σf Φ(A) − Ψ(f (B))
for all positive linear maps Φ, Ψ such that Φ(I) − Ψ(I) = I and for all selfadjoint
operators A and B with σ(A), σ(B) and σ(Φ(A) − Ψ(B)) ⊂ J. The invertibility of
Φ(I) guarantees the formulation of (3.3). In this case, we have
Φ(f (A)) ≤ Φ(I) σf Φ(A).
In fact, in the Stinespring decomposition theorem Φ(X) = C ∗ φ(X)C, we have the
polar decomposition C = V |C| such that |C| is invertible, because C ∗ C = Φ(I) =
I + Ψ(I) > 0. Since V ∗ V = I, it follows that
Φ(f (A)) = |C|V ∗ f (φ(A))V |C| ≤ |C|f (V ∗ φ(A)V )|C|
= |C|f (|C|−1 C ∗ φ(A)C|C|−1 )|C| = Φ(I) σf Φ(A).
If moreover C is invertible, then we have Φ(f (A)) = Φ(I) σf Φ(A), and hence,
(3.4)
see [6].
f (Φ(A) − Ψ(B)) ≤ Φ(f (A)) − Ψ(f (B)),
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Bounds for an Operator Concave Function
Based on the external version (3.4) of the Davis-Choi-Jensen inequality, we have
the following bounds for the difference of the operator concavity.
Theorem 3.1. Let Φ, Ψ, Φ′ and Ψ′ be positive linear maps from B(H) to B(K)
such that Φ(I) − Ψ(I) = I and Φ′ (I) − Ψ′ (I) = I and Φ is (β, α)-dominant by Φ′
and Ψ is (α, β)-dominant by Ψ′ . If a real-valued function f is operator concave on an
interval J, then
β (Φ′ (f (A)) − Ψ′ (f (B)) − f (Φ′ (A) − Ψ′ (B)))
≤ Φ(f (A)) − Ψ(f (B)) − f (Φ(A) − Ψ(B))
≤ α (Φ′ (f (A)) − Ψ′ (f (B)) − f (Φ′ (A) − Ψ′ (B)))
for all selfadjoint operators A and B with σ(A), σ(B), σ(Φ(A)−Ψ(B)) and σ(Φ′ (A)−
Ψ′ (B)) ⊂ J.
Proof. Put Φ1 (X) = αX. Since Φ is α-lower dominant by Φ′ and Ψ is α-upper
dominant by Ψ′ , and (Φ − αΦ′ )(I) + (αΨ′ − Ψ)(I) + Φ1 (I) = I, it follows from the
operator concavity of f that
f (Φ(A) − Ψ(B)) = f ((Φ − αΦ′ )(A) + (αΨ′ − Ψ)(B) + Φ1 (Φ′ (A) − Ψ′ (B)))
≥ (Φ − αΦ′ )(f (A)) + (αΨ′ − Ψ)(f (B)) + Φ1 (f (Φ′ (A) − Ψ′ (B)))
= Φ(f (A)) − Ψ(f (B)) − α (f (Φ′ (A) − Ψ′ (B)) − (Φ′ (f (A)) − Ψ′ (f (B))))
for all selfadjoint operators A and B with σ(A), σ(B), σ(Φ(A)−Ψ(B)) and σ(Φ′ (A)−
Ψ′ (B)) ⊂ J. This fact implies the second half of Theorem 3.1. Similarly, we obtain
the first half of Theorem 3.1.
Finally, we show an application of Theorem 3.1. Put positive linear maps Φ, Ψ,
Φ′ and Ψ′ : B(H) ⊕ · · · ⊕ B(H) 7→ B(H) as follows:
Pn
n
n
X
X
1 + i=1 pi
Φ(A1 ⊕ · · · ⊕ An ) =
pi Ai .
Ai and Ψ(A1 ⊕ · · · ⊕ An ) =
n
i=1
i=1
P
n
n
X
X
1 + ni=1 qi
′
′
qi Ai ,
Ai and Ψ (A1 ⊕ · · · ⊕ An ) =
Φ (A1 ⊕ · · · ⊕ An ) =
n
i=1
i=1
where pi , qi > 0 for i = 1, . . . , n. Then it P
follows that Φ(I) − Ψ(I) = I and Φ′ (I) −
1+ Pn
pi
pi
′
Ψ (I) = I. If we put α = max{ qi } and 1+ i=1
n
qi > α, then Φ is α-lower dominant of
i=1
1+
Pn
p
i
< β, then
Φ′ and Ψ is α-upper dominant of Ψ′ . If we put β = min{ pqii } and 1+Pi=1
n
i=1 qi
′
′
Φ is β-upper dominant of Φ and Ψ is β-lower dominant of Ψ . Hence, by Theorem 3.1,
we obtain the following external version of Corollary 2.2:
Corollary 3.2. Let f be operator convex on an interval J and A and Bi (i =
Pn
Pn
1, . . . , n) selfadjoint operators with σ(A), σ(Bi ) and σ((1 + i=1 pi )A − i=1 pi Bi ) ⊂
Electronic Journal of Linear Algebra ISSN 1081-3810
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Sever S. Dragomir, Jun Ichi Fujii, and Yuki Seo
J. Let α = max{ pqii } and β = min{ pqii }. If β >
β
f
(1 +
n
X
qi )A −
≤f
(1 +
pi )A −
P
1+ P pi
1+ qi
f
p i Bi
!
!
− ((1 +
n
X
then
n
X
qi )f (A) −
n
X
qi f (Bi ))
i=1
i=1
− ((1 +
!
pi )f (A) −
n
X
pi f (Bi ))
i=1
i=1
> α, then
(1 +
n
X
pi )A −
(1 +
n
X
p i Bi
i=1
i=1
≤α f
n
X
i=1
i=1
and if
qi Bi
i=1
i=1
n
X
n
X
P
1+ P pi
1+ qi ,
n
X
i=1
qi )A −
n
X
i=1
!
− ((1 +
qi Bi
n
X
pi )f (A) −
− ((1 +
pi f (Bi ))
i=1
i=1
!
n
X
n
X
i=1
qi )f (A) −
n
X
i=1
!
qi f (Bi )) .
REFERENCES
[1] M.D. Choi. A Schwarz inequality for positive linear maps on C∗ -algebras. Illinois Journal of
Mathematics, 18:565–574, 1974.
[2] C. Davis. A Schwartz inequality for convex operator functions. Proceedings of the American
Mathematical Society, 8:42–44, 1957.
[3] S.S. Dragomir. Bounds for the normalised Jensen functional. Bulletin of the Australian Mathematical Society, 74:471–478, 2006.
[4] S.S. Dragomir. Some inequalities of Jensen type for operator convex functions in Hilbert spaces.
Preprint, Reseach Group in Mathematical Inequalities and Applications, Reseach Report
Collection, 15:Article 40, 2012.
[5] J.I. Fujii. An external version of the Jensen operator inequality. Scientiae Mathematicae
Japonicae, 73:125–128, 2011.
[6] J.I. Fujii, J. Pečarić, and Y. Seo. The Jensen inequality in an external formula. Journal of
Mathematical Inequalities, 6:473–480, 2012.
[7] T. Furuta, J. Mićić Hot, J. Pečarić, and Y. Seo. Mond-Pečarić Method in Operator Inequalities.
Monographs in Inequalities, Vol. 1, Element, Zagreb, 2005.
[8] F. Kubo and T. Ando. Means of positive linear operators. Mathematische Annalen, 246:205–224,
1980.
[9] B. Mond and J. Pečarić. Remarks on Jensen’s inequality for operator convex functions. Annales
Universitatis Mariae Curie-Sklodowska. Sectio A. Mathematica, 47:96–103, 1993.
[10] W.F. Stinespring. Positive functions on C∗ -algebras. Proceedings of the American Mathematical
Society, 6:211–216, 1955.