Filomat 31:8 (2017), 2355–2364
DOI 10.2298/FIL1708355Y
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Further Generalizations of Some Operator Inequalities
Involving Positive Linear Map
Changsen Yanga , Chaojun Yanga
a Henan
Engineering Laboratory for Big Data Statistical Analysis and Optimal Control; College of Mathematics and Information Science, Henan
Normal University, Xinxiang 453007, Henan, P.R.China.
Abstract. We obtain a generalized conclusion based on an α-geometric mean inequality. The conclusion
is presented as follows: If m1 , M1 , m2 , M2 are positive real numbers, 0 < m1 ≤ A ≤ M1 and 0 < m2 ≤ B ≤ M2
for m1 < M1 and m2 < M2 , then for every unital positive linear map Φ and α ∈ (0, 1], the operator inequality
below holds:
p
(M1 +m1 )2 ((M1 +m1 )−1 (M2 +m2 ))2α )
Φp (A]α B), p ≥ 2.
(Φ(A)]α Φ(B))p ≤ 161
(m M )α (m M )1−α
2
2
1
1
Likewise, we give a second powering of the Diaz-Metcalf type inequality. Finally, we present p−th powering
of some reversed inequalities for n operators related to Karcher mean and power mean involving positive
linear maps.
1. Introduction
Let B(H) stand for the C∗ -algebra of all bounded linear operators acting on a complex Hilbert space H
and I denote the identity operator. k·k denote the operator norm. An operator A is said to be positive if
hAx, xi ≥ 0 for all x ∈ H and we write A ≥ 0. We identify T ≥ S(the same as S ≤ T) with T − S ≥ 0. A positive
1
invertible operator T is denoted by T > 0. The absolute value of T is denoted by |T| = (T∗ T) 2 , where T∗
stands for the adjoint of T.
A linear map Φ : B(H) → B(H) is called positive (strictly positive) if Φ(A) ≥ 0 (Φ(A) > 0) whenever
A ≥ 0 (A > 0), and Φ is said to be unital if Φ(I) = I. Take A, B > 0 and α ∈ [0, 1], the α-geometric mean A]α B
−1
−1
1
1
is defined by A]α B = A 2 (A 2 BA 2 )α A 2 , when α = 12 , A] 1 B = A]B is said to be the geometric mean between
2
A and B.
The geometric mean A]B of two positive operators A, B ∈ B(H) is characterized by Ando [2]
(
"
#
)
A X
∗
A]B = max X = X ∈ B(H) :
≥0 .
X B
2010 Mathematics Subject Classification. Primary 47A63 ; Secondary 46B10
Keywords. Positive linear map; α−geometric mean; operator inequality
Received: 25 November 2015; Accepted: 20 February 2016
Communicated by Dragan S. Djordjević
Research supported National Natural Science Foundation of China with grant (no. 11271112,11201127) and Technology and the
Innovation Team in Henan Province (NO.14IRTSTHN023).
Email addresses: [email protected] (Changsen Yang), [email protected] (Chaojun Yang)
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
2356
The definition above reveals the maximal characterization of geometric mean [16].
L The 2 × 2 operator
matrix mentioned in this work is naturally understood as an operator acting on H
H.
In 1948, Kantorovich [9] introduced the famous Kantorovich inequality. In 1990, Marshall and Olkin
[13] established an operator Kantorovich inequality. It is well known that tα is operator monotone function
on [0, ∞) if and only if α ∈ [0, 1]. Since t2 is not an operator monotone function, we can not obtain A2 ≥ B2
directly by A ≥ B ≥ 0. In 2013, Lin [12] proved that the operator Kantorovich inequality is order preserving
under squaring. In 2011, Seo gave an α-geometric mean inequality for positive linear map as follows:
Theorem 1.1. [17] Let Φ : B(H) → B(K ) be a unital positive linear map and let A and B be positive operators such
that 0 < m21 ≤ A ≤ M21 and 0 < m22 ≤ B ≤ M22 for some positive real numbers m1 < M1 and m2 < M2 . Then for
α ∈ [0, 1]
Φ(A)]α Φ(B) ≤ K(m, M, α)−1 Φ(A]α B),
(1)
m2 2
2 2
) = m, ( M
where we suppose ( M
m1 ) = M and the generalized Kantorovich constant K(M, m, α) [7] is defined by
1
K(m, M, α) =
mMα − Mmα α − 1 Mα − mα α
(α − 1)(M − m) α mMα − Mmα
for any real number α ∈ R.
Motived by Lin’s idea [12], Fu obtained a second powering of the operator inequality (1):
Theorem 1.2. [6] Let Φ : B(H) → B(K ) be a unital positive linear map and let A and B be positive operators such
that 0 < m1 ≤ A ≤ M1 and 0 < m2 ≤ B ≤ M2 for some positive real numbers m1 < M1 and m2 < M2 . Then for
α ∈ [0, 1]
(Φ(A)]α Φ(B))2 ≤
(M +m
1
2
−1
2α
1 ) ((M1 +m1 ) (M2 +m2 ))
4(m2 M2 )α (m1 M1 )1−α
2
Φ2 (A]α B).
(2)
Next we present the Diaz-Metcalf type inequality.
Theorem 1.3. [14] Let Φ be a positive linear map. If 0 < m21 ≤ A ≤ M21 and 0 < m22 ≤ B ≤ M22 for some positive real
numbers m1 ≤ M1 and m2 ≤ M2 , then the following inequality holds:
M2 m2
M2 m2
Φ(A) + Φ(B) ≤
+
Φ(A]B).
M1 m1
m1 M1
Pn
Let A1 , · · · , An > 0 and ω = (w1 , . . . , wn ) is a probability vector: i=1 wi = 1 and wi > 0 for i = 1, . . . , n. For
t ∈ (0, 1], the ω−weighted power mean Pt (ω; A1 , . . . , An ) (or Pt (ω; A))is defined as the unique solution of
X=
n
X
ωi (X]t Ai ).
i=1
−1
For t ∈ [−1, 0), we define Pt (ω; A1 , . . . , An ) = P−t (ω; A−1
, . . . , A−1
n ) . Next we bring some basic properties of
1
the power mean that is useful to obtain the results in this work, for more details about power mean, see
[11].
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
2357
Proposition 1.4. [11] The power mean satisfies the following properties:
−1
(i) (sel f duality) P−t (ω; A−1
, · · · , A−1
= Pt (ω; A1 , · · · , An ).
n )
1
Pn
Pn
−1
(ii) (AGH wei1hted mean inequalities) ( i=1 ωi A−1
≤ Pt (ω; A1 , · · · , An ) ≤ i=1 ωi Ai .
i )
(iii) t ∈ (0, 1], Φ(Pt (ω; A)) ≤ Pt (ω; Φ(A)) f or any positive unital linear map Φ; t ∈ [−1, 0),
Φ(Pt (ω; A)) ≥ Pt (ω; Φ(A)) f or any strictly positive unital linear map Φ.
(iv) Pt (ω; A) ≤ Pt (ω; B) i f Ai ≤ Bi f or all i = 1, 2, · · · , n.
The ω−weighted Karcher mean Λ(ω; A1 , . . . , An )(orΛ(ω; A)) of A1 , · · · , An > 0 is defined to be the unique
positive definite solution of equation
n
X
1
1
ωi lo1(X− 2 Ai X− 2 ) = 0,
i=1
where ω = (w1 , . . . , wn ) is a probability vector. Next we cite some basic properties of the Karcher mean as
follows, for more details about Karcher mean, see [11].
Proposition 1.5. [11] The Karcher mean satisfies the following properties:
−1
(i) (sel f duality) Λ(ω; A−1
, · · · , A−1
= Λ(ω; A1 , · · · , An ).
n )
1
Pn
Pn
−1
≤ Λ(ω; A1 , · · · , An ) ≤ i=1 ωi Ai .
(ii) (AGH wei1hted mean inequalities) ( i=1 ωi A−1
i )
(iii) Φ(Λ(ω; A)) ≤ Λ(ω; Φ(A)) f or any positive unital linear map Φ.
(iv) (monotonicity) I f Bi ≤ Ai f or all 1 ≤ i ≤ n, then Λ(ω; B) ≤ Λ(ω; A).
As mentioned in the abstract, we shall give a further generalization of Theorem 1.2 and Diaz-Metcalf type
inequality in the following section, along with presenting p−th powering of some reversed inequalities for
n operators related to Karcher mean and power mean involving positive linear maps.
2. Main Results
Before giving our main results, let us first consider the following lemmas.
Lemma 2.1. ( Choi inequality.) [5, 7] Let Φ be a unital positive linear map, then
(C1 ) when A > 0 and −1 ≤ p ≤ 0, then Φ(A)p ≤ Φ(Ap ), in particular,Φ(A)−1 ≤ Φ(A−1 );
(C2 ) when A ≥ 0 and 0 ≤ p ≤ 1, then Φ(A)p ≥ Φ(Ap );
(C3 ) when A ≥ 0 and 1 ≤ p ≤ 2, then Φ(A)p ≤ Φ(Ap ).
Lemma 2.2. [10] Let Φ be a unital positive linear map and A, B be positive operators. Then for α ∈ [0, 1]
Φ(A]α B) ≤ Φ(A)]α Φ(B).
Lemma 2.3. [4] Let A, B ≥ 0. Then the following norm inequality holds:
kABk≤
1
kA + Bk2 .
4
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
2358
Lemma 2.4. [3] Let A, B ≥ 0. Then for 1 ≤ r < +∞,
kAr + Br k≤ k(A + B)r k.
Now we are going to present the first main theorem of this paper.
Theorem 2.5. Let Φ : B(H) → B(K ) be a unital positive linear map and let A and B be positive operators such that
0 < m1 ≤ A ≤ M1 and 0 < m2 ≤ B ≤ M2 for some positive real numbers m1 < M1 and m2 < M2 . Then for α ∈ [0, 1],
p≥2
n
2
−1
2α op
1 (M1 +m1 ) ((M1 +m1 ) (M2 +m2 ))
(3)
(Φ(A)]α Φ(B))p ≤ 16
Φp (A]α B)
(m M )α (m M )1−α
2
2
1
1
Proof. The desired inequality is equivalent to
p
p
k(Φ(A)]α Φ(B)) 2 Φ− 2 (A]α B)k≤
1
4
n (M +m
2
−1
2α
1 ) ((M1 +m1 ) (M2 +m2 ))
(m2 M2 )α (m1 M1 )1−α
1
o p2
.
(4)
Note that
(M1 − A)(m1 − A)A−1 ≤ 0,
then
M1 m1 A−1 + A ≤ M1 + m1 ,
and therefore
M1 m1 Φ(A−1 ) + Φ(A) ≤ M1 + m1 .
(5)
Likewise, we can get
M2 m2 Φ(B−1 ) + Φ(B) ≤ M2 + m2 .
(6)
Hence, by the property of weighted geometric mean, through (5) and (6), we obtain
(M1 m1 Φ(A−1 ) + Φ(A))]α (M2 m2 Φ(B−1 ) + Φ(B)) ≤ (M1 + m1 )]α (M2 + m2 ).
Using the subadditivity of weighted geometric mean, we get
(m2 M2 )α (m1 M1 )1−α Φ−1 (A]α B) + Φ(A)]α Φ(B)
≤ (m2 M2 )α (m1 M1 )1−α (Φ(A−1 ]α B−1 )) + Φ(A)]α Φ(B)
≤ (m2 M2 )α (m1 M1 )1−α (Φ(A−1 )]α Φ(B−1 )) + Φ(A)]α Φ(B)
= (m1 M1 Φ(A−1 ))]α (m2 M2 Φ(B−1 )) + Φ(A)]α Φ(B)
≤ (m1 M1 Φ(A−1 ) + Φ(A))]α (m2 M2 Φ(B−1 ) + Φ(B)),
where the first inequality is obtained by Lemma 2.1, the second one is obtained by Lemma 2.2.
Since
αp
p
k(Φ(A)]α Φ(B)) 2 (m2 M2 ) 2 (m1 M1 )
p
αp
(1−α)p
2
p
Φ− 2 (A]α B)k
≤ 14 k(Φ(A)]α Φ(B)) 2 + (m2 M2 ) 2 (m1 M1 )
(1−α)p
2
p
Φ− 2 (A]α B)k2
p
≤ 14 k(Φ(A)]α Φ(B) + (m2 M2 )α (m1 M1 )1−α Φ−1 (A]α B)) 2 k2
= 14 kΦ(A)]α Φ(B) + (m2 M2 )α (m1 M1 )1−α Φ−1 (A]α B)kp
≤ 41 k(Φ(A) + m1 M1 Φ(A−1 ))]α (Φ(B) + m2 M2 Φ(B−1 ))kp
≤ 14 k(M1 + m1 )]α (M2 + m2 )kp
=
((M1 +m1 )((M1 +m1 )−1 (M2 +m2 ))α )p
,
4
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
2359
where the first inequality is obtained by Lemma 2.3, the second one is obtained by Lemma 2.4.,
so we have
p
p
k(Φ(A)]α Φ(B)) 2 Φ− 2 (A]α B)k
≤
((M1 +m1 )((M1 +m1 )−1 (M2 +m2 ))α )p
4(m2 M2 )
=
1
4
αp
2
(m1 M1 )
(1−α)p
2
n (M +m )2 ((M
−1
2α
1 +m1 ) (M2 +m2 ))
(m2 M2 )α (m1 M1 )1−α
1
1
o p2
.
Hence the inequality (4) has been obtained.
Remark 2.6. Inequality (2) is a special case of Theorem 2.5 by taking p = 2.
Lemma 2.7. [1] Let A, B ≥ 0. Then
A]B ≤
A+B
.
2
Inspired by Theorem 2.5, we give a further generalization related to the Diaz-Metcalf type inequality as
follows.
Theorem 2.8. Let Φ be a unital positive linear map. If 0 < m21 ≤ A ≤ M21 and 0 < m22 ≤ B ≤ M22 for some positive
real numbers m1 ≤ M1 and m2 ≤ M2 , then the following inequality holds:
M 2 m2
M1 m1 Φ(A)
+ Φ(B)
p
≤
1
16
(M1 m1 (M22 +m22 )+M2 m2 (M21 +m21 ))2
√
2 M2 M1 m1 m2 M21 m21 M2 m2
p
Φp (A]B),
p ≥ 2.
(7)
Proof. Obviously (7) is equivalent to
M2 m2
k
Φ(A) + Φ(B)
M1 m1
2p
p
−2
Φ

p
2 2 2
2
2
2


))
+
m
(M
m
(M
+
m
)
+
M
m
(M
1
1
2
2
1


2
2
1
1
(A]B)k≤ 
.
√



2
2

4
2 M2 M1 m1 m2 M1 m1 M2 m2
By the proof of (5), we have
M21 m21 Φ(A−1 ) + Φ(A) ≤ M21 + m21 ,
which equals to
M2 m2 m1 M1 Φ(A−1 ) +
M2 m2
M1 m1 Φ(A)
≤
M 2 m2
2
M1 m1 (M1
+ m21 ).
(8)
Similarly, we have
M22 m22 Φ(B−1 ) + Φ(B) ≤ M22 + m22 .
(9)
By (8) and (9) we have
√
M2 m2
2 M2 M1 m1 m2 M2 m2 Φ(A]B)−1 + ( M
Φ(A) + Φ(B))
1 m1
√
M2 m2
≤ 2 M2 M1 m1 m2 M2 m2 Φ(A−1 ]B−1 ) + ( M
Φ(A) + Φ(B))
1 m1
√
M2 m2
≤ 2 M2 M1 m1 m2 M2 m2 Φ(A−1 )]Φ(B−1 ) + ( M
Φ(A) + Φ(B))
1 m1
≤ M2 M1 m1 m2 Φ(A−1 ) + M22 m22 Φ(B−1 ) +
≤
M 2 m2
2
M1 m1 (M1
+ m21 ) + M22 + m22 ,
M2 m2
M1 m1 Φ(A)
+ Φ(B)
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
2360
where the first inequality is obtained by Lemma 2.1, the second one is obtained by Lemma 2.2, the third
one is obtained by Lemma 2.7, the last one is the result of combining (8) and (9).
Since
2p √
p
p
2 m2
(2 M2 M1 m1 m2 M2 m2 ) 2 Φ− 2 (A]B)k
k M
M1 m1 Φ(A) + Φ(B)
2p
√
p
p
2 m2
≤ 14 k M
+ (2 M2 M1 m1 m2 M2 m2 ) 2 Φ− 2 (A]B)k2
Φ(A)
+
Φ(B)
M 1 m1
p2
√
−1
2 m2
M
M
m
m
M
m
Φ
(A]B)
≤ 14 k M
k2
Φ(A)
+
Φ(B)
+
2
2
1
1
2
2
2
M 1 m1
√
−1
p
2 m2
= 14 k M
M1 m1 Φ(A) + Φ(B) + 2 M2 M1 m1 m2 M2 m2 Φ (A]B)k
p
M2 m2 (M21 +m21 )+M1 m1 (M22 +m22 )
≤ 41
,
M 1 m1
we have
M2 m2
k
Φ(A) + Φ(B)
M1 m1
2p
p
−2
Φ

p
2
2
2
2 2 2

1
 (M1 m1 (M2 + m2 ) + M2 m2 (M1 + m1 )) 

(A]B)k≤ 
.
√


2 2


4
2 M2 M1 m1 m2 M m M2 m2
1
1
Corollary 2.9. In Theorem 2.8, put p = 2, we get

2
2 
2
2
2
2 2

M2 m2
 (M1 m1 (M2 + m2 ) + M2 m2 (M1 + m1 )) 
 2
Φ(A) + Φ(B) ≤ 
Φ (A]B),
√





M1 m1
8 M2 M1 m1 m2 M21 m21 M2 m2
which can be seen as a squared operator inequality related to Diaz-Metcalf type inequality.
Lemma 2.10. [8] For any bounded operator X,
"
#
X
≥0
tI
tI
|X| ≤ tI ⇔ kXk≤ t ⇔ ∗
X
(t ≥ 0)
Theorem 2.11. Let A and B be positive operators on a Hilbert space H with 0 < m1 ≤ A ≤ M1 and 0 < m2 ≤ B ≤ M2 ,
Φ be a unital positive linear map on B(H). Then for 0 ≤ α ≤ 1 and p ≥ 2,
p
p
p
p
(Φ(A)]α Φ(B)) 2 Φ− 2 (A]α B) + Φ− 2 (A]α B)(Φ(A)]α Φ(B)) 2
p
n
2
−1
2α o 2
1 (M1 +m1 ) ((M1 +m1 ) (M2 +m2 ))
≤2
.
(m2 M2 )α (m1 M1 )1−α
Proof. By (4) and Lemma 2.10, we obtain
 n
op
 1 (M1 +m1 )2 ((M1 +m1 )−1 (M2 +m2 ))2α 2 I
 4
α
1−α
(m2 M2 ) (m1 M1 )


p
p
−2
Φ (A] B)(Φ(A)] Φ(B)) 2
α
and
α
 n
op
 1 (M1 +m1 )2 ((M1 +m1 )−1 (M2 +m2 ))2α 2 I
 4
(m2 M2 )α (m1 M1 )1−α


p
p
(Φ(A)] Φ(B)) 2 Φ− 2 (A] B)
α
α
(10)
p
1
4
p
(Φ(A)]α Φ(B)) 2 Φ− 2 (A]α B)
n (M +m )2 ((M +m )−1 (M +m ))2α o p2
1
1
1
1
2
(m2 M2 )α (m1 M1 )1−α
2
p
1
4
p
Φ− 2 (A]α B)(Φ(A)]α Φ(B)) 2
n (M +m )2 ((M +m )−1 (M +m ))2α o p2
1
1
1
1
2
(m2 M2 )α (m1 M1 )1−α
2
Summing up these two operator matrices above, and putting
(
)p
1 (M1 + m1 )2 ((M1 + m1 )−1 (M2 + m2 ))2α 2
t=
,
4
(m2 M2 )α (m1 M1 )1−α



 ≥ 0

I



 ≥ 0

I
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
p
p
2361
p
p
X = Φ− 2 (A]α B)(Φ(A)]α Φ(B)) 2 + (Φ(A)]α Φ(B)) 2 Φ− 2 (A]α B),
we have
"
2tI
X
p
p
#
X
≥0
2tI
p
p
Since Φ− 2 (A]α B)(Φ(A)]α Φ(B)) 2 + (Φ(A)]α Φ(B)) 2 Φ− 2 (A]α B) is self-adjoint, (10) follows from the maximal
characterization of geometric mean.
Theorem 2.12. Let Φ be a unital positive linear map. If 0 < m21 ≤ A ≤ M21 and 0 < m22 ≤ B ≤ M22 for some positive
real numbers m1 ≤ M1 and m2 ≤ M2 , p ≥ 2, then the following inequalities holds:
p
p
p
p
(Φ(A)]α Φ(B)) 2 Φ− 2 (A]α B) + Φ− 2 (A]α B)(Φ(A)]α Φ(B)) 2
2p
2
2
2
2 2
1 (M1 m√1 (M2 +m2 )+M2 m2 (M1 +m1 ))
≤2
2 M M m m M2 m2 M m
2
1
1
2
1
1
2
(11)
2
Proof. By the method of proving Theorem 2.11, we can easily get (11).
Next we present a p−th powering of a reversed HM-PM inequality for n operators.
Theorem 2.13. Let 0 < m ≤ Ai ≤ M for i = 1, · · · n, M ≥ m, ω = (ω1 , · · · , ωn ) be a probability vector,
t ∈ [−1, 0) ∪ (0, 1]. Then we have
Pt (ω; A1 , · · · , An )p ≤
1
16
n (M+m)2 op P
n
)−p ,
( ωi A−1
Mm
i
p ≥ 2.
(12)
i=1
Proof. The inequality (12) is equivalent to
p
2
kPt (ω; A1 , · · · , An ) (
n
X
(
)p
1 (M + m)2 2
.
4
Mm
p
2
ωi A−1
i ) k≤
i=1
Since (5) implies
Mm(
n
P
i=1
ωi A−1
)+
i
n
P
ωi Ai ≤ M + m.
(13)
i=1
Then by Proposition 1.4, we have
Pt (ω; A1 , · · · , An ) + Mm(
n
P
i=1
≤
n
P
ωi Ai + Mm(
i=1
n
P
i=1
ωi A−1
)
i
ωi A−1
)
i
≤ M + m.
Thus
p
p
p
kPt ω; A1 , · · · , An ) 2 (Mm) 2 (ωi A−1
)2 k
i
p
p
p
≤ 14 kPt (ω; A1 , · · · , An ) 2 + (Mm) 2 (ωi A−1
) 2 k2
i
p
≤ 41 k(Pt (ω; A1 , · · · , An ) + Mm(ωi A−1
)) 2 k2
i
= 14 kPt (ω; A1 , · · · , An ) + Mm(ωi A−1
)kp
i
≤ 14 (M + m)p .
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
Therefore
p
2
kPt (ω; A1 , · · · , An ) (
n
X
p
2
ωi A−1
i ) k≤
i=1
2362
(
)p
1 (M + m)2 2
.
4
Mm
From [11] we know, power mean is increasing from [−1, 0) ∪ (0, 1]. Furthermore, lim Pt (ω; Φ(A)) =
t→0
Λ(ω; Φ(A)). Therefore in Theorem 2.13 put t → 0, we obtain:
Corollary 2.14. Let 0 < m ≤ Ai ≤ M for i = 1, · · · n, M ≥ m, ω = (ω1 , · · · , ωn ) be a probability vector. Then we have
Λ(ω; A1 , · · · , An )p ≤
1
16
n (M+m)2 op P
n
)−p ,
( ωi A−1
Mm
i
p ≥ 2.
(14)
i=1
Lemma 2.15 presented a p-th powering of a reversed HM-KM inequality for n operators.
Lemma 2.15. [7] (L-H inequality) If 0 ≤ α ≤ 1, A, B ≥ 0 and A ≥ B, then Aα ≥ Bα .
Lemma 2.16. Let 0 < m ≤ Ai ≤ M for i = 1, · · · n, M ≥ m, ω = (ω1 , · · · , ωn ) be a probability vector. Then we obtain
n
P
ωi Ai ≤
i=1
Proof. First we proof (
n
P
ω i Ai ) 2 ≤ (
i=1
n
(m+M)2 P
ωi A−1
)−1 .
4mM (
i
i=1
n
(m+M)2 2 P
)
(
ωi A−1
)−2 . This inequality
4mM
i
i=1
n
n
X
X
(M + m)2
k
ωi Ai
.
ωi A−1
i k≤
4Mm
i=1
i=1
equals to
Now by
k(
n
P
ωi Ai )Mm(
i=1
n
P
i=1
≤ 14 k
n
P
)k
ωi A−1
i
ωi Ai + Mm(
i=1
n
P
i=1
ωi A−1
)k2
i
≤ 41 (M + m)2 ,
where the second inequality is obtained by (13). Hence we can get desired inequality by Lemma 2.15.
We remark that the result of Lemma 2.16 is a special case of Theorem 1 in [15].
Theorem 2.17. Let Φ be a unital positive linear map, 0 < m ≤ Ai ≤ M for i = 1, · · · n, M ≥ m. ω = (ω1 , · · · , ωn ) be
a probability vector, t ∈ (0, 1]. Then we have
Pt (ω; Φ(A)) ≤
(m+M)2
4mM Φ(Λ(ω; A)).
(15)
Proof. By Proposition 1.4, 1.5 and Lemma 2.16 we get
Pt (ω; Φ(A)) ≤
n
X
i=1
n
n
X
X
(m + M)2
(m + M)2
−1
ωi Φ(Ai ) = Φ(
ωi Ai ) ≤
Φ((
ωi A−1
Φ(Λ(ω; A)).
i ) )≤
4mM
4mM
i=1
i=1
From [11] we know Pt (ω; A) ≥ Λ(ω; A) when t ∈ (0, 1], and by Proposition 1.4, 1.5 we have Pt (ω; Φ(A)) ≥
Λ(ω; Φ(A)) ≥ Φ(Λ(ω; A)) . Thus (15) can be viewed as a reversed PM-KM inequality involving positive
unital linear maps.
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
2363
Corollary 2.18. Under the same conditions as in Theorem 2.13, one can obtain
Φ(Pt (ω; A)) ≤
(m+M)2
4mM Λ(ω; Φ(A)).
(16)
Proof. By Proposition 1.4 and 1.5, we obtain
Φ(Pt (ω; A)) ≤ Pt (ω; Φ(A))) ≤
(m + M)2
(m + M)2
Φ(Λ(ω; A)) ≤
Λ(ω; Φ(A)).
4mM
4mM
Next we present p-th powering of (14) involving positive unital linear maps.
Theorem 2.19. Let Φ be a unital positive linear map, 0 < m ≤ Ai ≤ M for i = 1, · · · n, M ≥ m. ω = (ω1 , · · · , ωn ) be
a probability vector, t ∈ (0, 1], p ≥ 2. Then we have
n
2 op
1 (M+m)
(17)
Φ(Λ(ω; A))p .
Pt (ω; Φ(A))p ≤ 16
Mm
Proof. The inequality (17) is equivalent to
p
2
kPt (ω; Φ(A)) Φ(Λ(ω; A))
p
−2
(
)p
1 (M + m)2 2
k≤
.
4
Mm
By (13), we obtain
Pt (ω; Φ(A)) + MmΦ(Λ(ω; A))−1
≤ Pt (ω; Φ(A)) + MmΦ(Λ(ω; A)−1 )
≤(
n
P
ωi Φ(Ai )) + MmΦ(
i=1
n
P
i=1
ωi A−1
)
i
≤ M + m.
Then
p
p
p
kPt (ω; Φ(A)) 2 (Mm) 2 Φ(Λ(ω; A))− 2 k
p
p
p
≤ 14 kPt (ω; Φ(A)) 2 + (Mm) 2 Φ(Λ(ω; A))− 2 k2
≤ 14 kPt (ω; Φ(A)) + MmΦ(Λ(ω; A))−1 kp
≤ 14 (M + m)p .
Therefore we have the desired inequality.
Theorem 2.20. Let Φ be a strictly unital positive linear map, 0 < m ≤ Ai ≤ M for i = 1, · · · n, M ≥ m.
ω = (ω1 , · · · , ωn ) be a probability vector, t ∈ [−1, 0). Then we have
Λ(ω; Φ(A)) ≤
(m+M)2
4mM Φ(Pt (ω; A)).
(18)
Proof. By Proposition 1.4, Proposition 1.5 and Lemma 2.16 we get
Λ(ω; Φ(A)) ≤
n
X
i=1
n
n
X
X
(m + M)2
(m + M)2
−1
ωi Φ(Ai ) = Φ(
ωi Ai ) ≤
Φ((
ωi A−1
Φ(Pt (ω; A)).
i ) )≤
4mM
4mM
i=1
i=1
From [11] we know Λ(ω; A) ≥ Pt (ω; A) when t ∈ [−1, 0), and by Proposition 1.4, 1.5 we have Λ(ω; Φ(A)) ≥
Φ(Λ(ω; A)) ≥ Φ(Pt (ω; A)) . Thus (18) can be viewed as a reversed PM-KM inequality as well.
Next we give a p-th powering of (18).
C. Yang, C. Yang / Filomat 31:8 (2017), 2355–2364
2364
Theorem 2.21. Let Φ be a strictly unital positive linear map, 0 < m ≤ Ai ≤ M for i = 1, · · · n, M ≥ m.
ω = (ω1 , · · · , ωn ) be a probability vector, t ∈ [−1, 0), p ≥ 2. Then we have
n
2 op
1 (M+m)
(19)
Λ(ω; Φ(A))p ≤ 16
Φ(Pt (ω; A))p .
Mm
Proof. Since
Λ(ω; Φ(A)) + Φ(Pt (ω; A))−1
≤ Λ(ω; Φ(A)) + Φ(Pt (ω; A)−1 )
= Λ(ω; Φ(A)) + Φ(P−t (ω; A−1 ))
≤(
n
P
ωi Φ(Ai )) + MmΦ(
i=1
n
P
i=1
ωi A−1
)
i
≤ M + m,
so
p
p
p
kΛ(ω; Φ(A)) 2 (Mm) 2 Φ(Pt (ω; A))− 2 k
p
p
p
≤ 14 kΛ(ω; Φ(A)) 2 + (Mm) 2 Φ(Pt (ω; A))− 2 k2
≤ 14 kΛ(ω; Φ(A)) + MmΦ(Pt (ω; A))−1 kp
≤ 14 (M + m)p .
Therefore we have
p
2
kΛ(ω; Φ(A)) Φ(Pt (ω; A))
p
−2
(
)p
1 (M + m)2 2
,
k≤
4
Mm
which is equivalent (19).
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