Some operator Bellman type inequalities

SOME OPERATOR BELLMAN TYPE INEQUALITIES

arXiv:1504.08299v1 [math.FA] 30 Apr 2015

MOJTABA BAKHERAD1 AND ALI MORASSAEI2 Abstract. In this paper, we employ the Mond–Peˇcarić method to establish some reverses of the operator Bellman inequality under certain conditions. In particular, we show δIK +

n X j=1



ωj Φj ((IH − Aj )p ) ≥ 

n X j=1

p

ωj Φj (IH − Aj ) ,

where Aj (1 ≤ j ≤ n) are self-adjoint contraction operators with 0 ≤ mIH ≤ Aj ≤ M IH , Φj are unital positive linear maps on B(H ), ωj ∈ R+ (1 ≤ j ≤ n), p p p −(1−M)p p−1 −(1−m)(1−M)p 0 < p < 1 and δ = (1 − p) 1p (1−m)M−m . + (1−M)(1−m)M−m We also present some refinements of the operator Bellman inequality.

1. Introduction Let B(H ) denote the C ∗ -algebra of all bounded linear operators on a complex Hilbert space H with the identity IH . In the case when dimH = n, we identify B(H ) with the matrix algebra Mn (C) of all n × n matrices with entries in the complex field. An operator A ∈ B(H ) is called positive if hAx, xi ≥ 0 for all x ∈ H and in this case we write A ≥ 0. We write A > 0 if A is a positive invertible operator. The set of all positive invertible operators is denoted by B(H )+ . For self-adjoint operators A, B ∈ B(H ), we say A ≤ B if B − A ≥ 0. Also, an operator A ∈ B(H ) is said to be contraction, if A∗ A ≤ IH . The Gelfand map f (t) 7→ f (A) is an isometrical ∗-isomorphism between the C ∗ -algebra C(sp(A)) of continuous functions on the spectrum sp(A) of a self-adjoint operator A and the C ∗ -algebra generated by A and IH . If f, g ∈ C(sp(A)), then f (t) ≥ g(t) (t ∈ sp(A)) implies that f (A) ≥ g(A). Let f be a continuous real valued function defined on an interval J. It is called operator monotone if A ≤ B implies f (A) ≤ f (B) for all self-adjoint operators A, B ∈ B(H ) with spectra in J; see [5] and references therein for some recent results. It said to be operator concave if λf (A) + (1 − λ)f (B) ≤ f (λA + (1 − λ)B) 2010 Mathematics Subject Classification. Primary 47A63, Secondary 46L05, 47A60. Key words and phrases. Bellman inequality; operator mean; the Mond–Peˇcarić method; positive linear map. 1

2

M. BAKHERAD, A. MORASSAEI

for all self-adjoint operators A, B ∈ B(H ) with spectra in J and all λ ∈ [0, 1]. Every nonnegative continuous function f is operator monotone on [0, +∞) if and only if f is operator concave on [0, +∞); see [7, Theorem 8.1]. A map Φ : B(H ) −→ B(K ) is called positive if Φ(A) ≥ 0 whenever A ≥ 0, where K is a complex Hilbert space and is said to be unital if Φ(IH ) = IK . We denote by P[B(H ), B(K )] the set of all positive linear maps Φ : B(H ) → B(K ) and by PN [B(H ), B(K )] the set of all unital positive linear maps. The axiomatic theory for operator means of positive invertible operators have been developed by Kubo and Ando [9]. A binary operation σ on B(H )+ is called a connection, if the following conditions are satisfied: (i) A ≤ C and B ≤ D imply AσB ≤ CσD; (ii) An ↓ A and Bn ↓ B imply An σBn ↓ AσB, where An ↓ A means that A1 ≥ A2 ≥ · · · and An → A as n → ∞ in the strong operator topology; (iii) T ∗ (AσB)T ≤ (T ∗ AT )σ(T ∗BT ) (T ∈ B(H )). There exists an affine order isomorphism between the class of connections and the class of positive operator monotone functions f defined on (0, ∞) via f (t)IH = IH σf (tIH ) (t > 0).

1

1

1

1

In addition, Aσf B = A 2 f (A− 2 BA− 2 )A 2 for all A, B ∈

B(H )+ . The operator monotone function f is called the representing function of σf . A connection σf is a mean if it is normalized, i.e. IH σf IH = IH . The function f∇µ (t) = (1 − µ) + µt and f♯µ (t) = tµ on (0, ∞) for µ ∈ (0, 1) give the operator weighted arithmetic mean A∇µ B = (1− µ)A + µB and the operator weighted geoµ 1 1 1 1 metric mean A♯µ B = A 2 A− 2 BA− 2 A 2 , respectively. The case µ = 1/2, the operator weighted geometric mean gives rise to the so–called geometric mean A♯B.

Bellman [3] proved that if p is a positive integer and a, b, aj , bj (1 ≤ j ≤ n) are P P positive real numbers such that nj=1 apj ≤ ap and nj=1 bpj ≤ bp , then ap −

n X j=1

apj

!1/p

bp −

+

n X j=1

bpj

!1/p

≤

n X p (aj + bj )p (a + b) − j=1

!1/p

.

A multiplicative analogue of this inequality is due to J. Aczél. In 1956, Aczél [1] P proved that if aj , bj (1 ≤ j ≤ n) are positive real numbers such that a21 − nj=2 a2j > 0 P or b21 − nj=2 b2j > 0, then a21

−

n X j=2

a2j

!

b21

−

n X j=2

b2j

!

≤

a1 b1 −

n X j=2

aj bj

!2

.


3

Popoviciu [15] extended Aczél’s inequality by showing that ! ! !p n n n X X X p p p p a1 − aj b1 − bj ≤ a1 b1 − aj bj , j=2

where p ≥ 1 and ap1 −

Pn

j=2

j=2

apj > 0 or bp1 −

j=2

Pn

p j=2 bj

> 0.

During the last decades several generalizations, refinements and applications of the Bellman inequality in various settings have been given and some results related to integral inequalities are presented; see [4, 11, 12, 13, 16] and references therein. In [12] the authors showed an operator Bellman inequality as follows: Φ ((IH − A)p ∇λ (IH − B)p ) ≤ (Φ(IH − A∇λ B))p , whenever A, B are positive contraction operators, Φ is a unital positive linear map on B(H ) and 0 < p < 1. They also [11] showed the following generalization of the Bellman operator inequality ! ! n n X X IH − Aj σf p IH − Bj ≤ j=1

IH −

j=1

n X

Aj σf Bj

j=1

where Aj , Bj (1 ≤ j ≤ n) are positive operators such that

Pn

!!p

,

j=1 Aj ≤ IH ,

IH , σf is a mean with the representing function f and 0 < p < 1.

(1.1) Pn

j=1 Bj

≤

In this paper, we use the Mond–Peˇcarić method to present some reverses of the operator Bellman inequality under some mild conditions. We also show some refinements of (1.1). 2. Some reverses of the Bellman type operator inequality The operator Choi-Davis-Jensen inequality says that if f is an operator concave function on an interval J and Φ ∈ PN [B(H ), B(K )], then f (Φ(A)) ≥ Φ(f (A)) for all self-adjoint operators A with spectrum in J. The Mond–Peˇcarić method [7, Chapter 2] present that if f is a strictly concave differentiable function on an interval [m, M] with m < M and Φ ∈ PN [B(H ), B(K )], M f (m) − mf (M ) f (M ) − f (m) , νf = and γf = max µf = M −m M −m

f (t) :m≤t≤M µf t + ν f

, (2.1)

then γf Φ(f (A)) ≥ f (Φ(A)).

(2.2)

In inequality (2.2), if we put Φ(X) := Ψ(A)−1/2 Ψ(A1/2 XA1/2 )Ψ(A)−1/2 , where Ψ is an arbitrary unital positive linear map and take f to be the representing function

4


of an operator mean σf , then we reach the inequality f (t) max Ψ(Aσf B) ≥ Ψ(A)σf Ψ(B) m≤t≤M µf t + νf

(2.3)

whenever 0 < mA ≤ B ≤ MA. Finally, if we take Ψ in (2.3) to be the positive unital linear map defined on the P diagonal blocks of operators by Ψ(diag(A1 , · · · , An )) = n1 nj=1 Aj , then ! ! n n n X X X γf Aj σf Bj ≥ Aj σf Bj , (2.4) j=1

j=1

j=1

where γf is given by (2.1) and 0 < mAj ≤ Bj ≤ MAj (1 ≤ j ≤ n). In the following theorem we show a reversed operator Bellman type inequality. Theorem 2.1. Suppose that 0 < mAj ≤ Bj ≤ MAj (1 ≤ j ≤ n) and 0 < P P Pn m IH − γf j=1 Aj ≤ IH − γf nj=1 Bj ≤ M IH − γf nj=1 Aj for some positive real numbers m, M such that m < 1 < M, γf is given by (2.1), σf is an operator mean with the representing function f and p ∈ [0, 1]. Then γfp



IH −

n X j=1





Aj  σf IH −

n X j=1

Proof. By using (2.4) we have γf

n+1 X j=1

p



Bj  ≥ IH 

Xj σf Yj ≥ 

n+1 X j=1



p  n X Aj σf Bj  . − γf 



Xj  σf 

(2.5)

j=1

n+1 X j=1



Yj  ,

where 0 < mXj ≤ Yj ≤ MXj (1 ≤ j ≤ n + 1). If we take Xj = Aj , Yj = Bj (1 ≤ P P j ≤ n), Xn+1 = IH − nj=1 Aj ≥ 0 and Yn+1 = IH − nj=1 Bj ≥ 0, then         n n n n X X X X Bj  ≥ IH σf IH = IH , Aj  σf IH − Bj  + IH − Aj  σf  γf   j=1

j=1

j=1

j=1

whence



IH −

n X j=1





Aj  σf IH



    n n X X 1 Bj  ≥ − Bj  . IH −  Aj  σf  γ f j=1 j=1 j=1 n X

(2.6)

It follows from inequality (2.6) and the Löwner-Heinz inequality [7, Theorem 1.8] that 

IH −

n X j=1





Aj  σf IH

p

   p n n X X 1 − Bj  ≥  IH −  Aj  σf  Bj  . γ f j=1 j=1 j=1 n X




5

Lemma 2.2. Suppose that C, X ∈ B(H ) such that C is a contraction, 0 < mIH ≤ X ≤ MIH , f is a concave and operator monotone function on [m, M] and γf is given by (2.1). Then γf [C ∗ f (X)C + f (m)(IH − C ∗ C)] ≥ f (C ∗ XC).

(2.7)

1 2

Proof. Let D = (IH −C ∗ C) . Consider the positive unital linear map Φ

"

X

0

#!

0 Y C ∗ XC + D ∗ Y D (X, Y ∈ B(H )). Using inequality (2.2) and the operator monotonicity of f we have f (C ∗ XC) ≤ f (C ∗ XC + D ∗ mD) " #!! X 0 =f Φ 0 m " #!! f (X) 0 ≤ γf Φ (by (2.2)) 0 f (m) = γf [C ∗ f (X)C + D ∗ f (m)D] , whenever 0 < mIH ≤ X ≤ MIH .

Lemma 2.3. Let 0 < mA ≤ B ≤ MA with A contraction. Let σf be an operator mean with the representing function f and h be an operator monotone function on [0, +∞). Then γh h(f (m)) IH − A + (Aσhof B) ≥ h (Aσf B) ,

where µh =

h(f (M ))−h(f (m)) , νh f (M )−f (m)

=

f (M )h(f (m))−f (m)h(f (M )) f (M )−f (m)

and γh =

(2.8)

h(t) . f (m)≤t≤f (M ) µh t + νh max

Proof. It follows from f (m) ≤ f A−1/2 BA−1/2 ≤ f (M) and inequality (2.7) that

h (Aσf B) = h A1/2 f A−1/2 BA−1/2 A1/2 ≤ γh A1/2 h f A−1/2 BA−1/2 A1/2 + (IH − A)1/2 h(f (m)) (IH − A)1/2 (by (2.7))

= γh [(Aσhof B) + h(f (m)) (IH − A)] .

=

6


Applying the operator monotone function f (t) = (1 − λ) + λt (λ ∈ [0, 1]) in Theorem 2.1, due to f (t) (1 − λ) + λt = max =1 m≤t≤M µf t + νf m≤t≤M (1 − λ) + λt

γf = max

and using Lemma 2.3 for the special case h(t) = tp (p ∈ [0, 1]), due to γh =

pp (f (M) − f (m))(f (M)f (m)p − f (m)f (M)p )p−1 tp = f (m)≤t≤f (M ) µh t + νh (1 − p)p−1 (f (M)p − f (m)p )p max

we have the following result; see [7, p. 77]. Corollary 2.4 (A reverse operator Bellman Let 0 < mAj ≤ Bj ≤ inequality). P Pn Pn MAj (1 ≤ j ≤ n) and 0 < m IH − j=1 Aj ≤ IH − j=1 Bj ≤ M IH − nj=1 Aj for some positive real numbers m, M such that m < 1 < M and p ∈ [0, 1]. Then

  p       n n n n X X X X Bj  ≥ IH −  Aj σf Bj  , Aj  σ((1−λ)+λt)p IH − Aj  + IH − δ f (m)p  

where δ =

j=1

j=1

j=1

pp (f (M )−f (m))(f (M )f (m)p −f (m)f (M )p )p−1 (1−p)p−1 (f (M )p −f (m)p )p

j=1

and f (t) = (1 − λ) + λt (λ ∈ [0, 1]).

Proof. We have 



δ f (m)p 

n X j=1



≥ IH − 



≥ IH − 





Aj  + IH −

n X j=1

n X j=1



n X



j=1

Aj  ∇λ IH − p





Aj  σ((1−λ)+λt)p IH −

n X j=1

p

n X j=1



Bj  

Bj  (by (2.8))

Aj ∇λ Bj  (by (2.5)).

In [8], the authors showed another way to find a reverse Choi-Davis-Jensen inequality. If f is a strictly concave differentiable function on an interval [m, M] with m < M and Φ is a unital positive linear map, then βf IH + Φ(f (A)) ≥ f (Φ(A)),

(2.9)

where A ∈ B(H ) is a self-adjoint operator with spectrum in [m, M] and βf = maxm≤t≤M {f (t) − µf t − νf }. In inequality (2.9), if we put Ψ(X) := Ψ(A)−1/2 Ψ(A1/2 XA1/2 )Ψ(A)−1/2 , where Ψ is an arbitrary unital positive linear map and take f to be the representing function of an operator mean σf , then we reach the inequality βf Ψ(X) + Ψ(Xσf Y ) ≥ Ψ(X)σf Ψ(Y ),

(2.10)


7

where 0 < mX ≤ Y ≤ MX, σf is an operator mean with representing function f and βf = maxm≤t≤M {f (t) − µf t − νf } that is the unique solution of the equation f ′ (t) = µf , whenever µf =

f (M )−f (m) M −m

and νf =

M f (m)−mf (M ) . M −m

Applying (2.10) to the positive unital linear map defined on the diagonal blocks of P operators by Ψ(diag(X1 , · · · , Xn+1 )) = n1 n+1 j=1 Xj , we get ! ! n+1 n+1 n+1 n+1 X X X X βf Xj + Yj σf Xj ≥ Xj σf Yj , (2.11) j=1

j=1

j=1

j=1

where 0 < mXj ≤ Yj ≤ MXj (1 ≤ j ≤ n + 1), σf is an operator mean with representing function f and βf = maxm≤t≤M {f (t) − µf t − νf } . Now, we have the next result. Proposition 2.5. Suppose that 0 < mAj ≤ Bj ≤ MAj (1 ≤ j ≤ n) and 0 < Pn P P m IH − j=1 Aj ≤ IH − nj=1 Bj ≤ M IH − nj=1 Aj for some positive real

numbers m, M such that m < 1 < M and σf is an operator mean with representing function f . Then βf +

IH −

n X j=1

Aj

!

σf

IH −

n X

Bj

j=1

!!p

≥

IH −

n X

Aj σf Bj

j=1

!!p

,

whenever p ∈ [0, 1] and βf = maxm≤t≤M {f (t) − µf t − νf } . P Proof. If we take Xj = Aj , Yj = Bj (1 ≤ j ≤ n), Xn+1 = IH − nj=1 Aj and P Yn+1 = IH − nj=1 Bj in inequality (2.11), then we get ! ! !! n n n X X X βf + IH − Aj σf IH − Bj ≥ IH − Aj σf Bj . (2.12) j=1

j=1

j=1

By the operator monotonicity of h(t) = tp and (2.12) we reach the desired inequality. Corollary 2.6 (A reverse Aczél type inequality). Let 0 < mAj ≤ Bj ≤MAj (1 ≤ P P P j ≤ n), 0 < m IH − nj=1 Aj ≤ IH − nj=1 Bj ≤ M IH − nj=1 Aj for some positive real numbers m, M such that m < 1 < M. Then ! !!p n n X X ζ + IH − Aj ♯λ IH − Bj ≥ IH − j=1

where p, λ ∈ [0, 1] and ζ = (1 − p)

j=1

M p −mp p(M −m)

p p−1

−

M mp −mM p . M −m

n X j=1

Aj ♯λ Bj

!!p

,

8


Let Aj ∈ B(H ) (1 ≤ j ≤ n) be self-adjoint operators with sp(Aj ) ⊆ [m, M] for some scalars m < M, Φj be unital positive linear maps on B(H ), ω1 , · · · , ωn ∈ R+ P be any finite number of positive real numbers such that nj=1 ωj = 1 and f be a

strictly concave differentiable function. If we take the positive unital linear map P Φ(diag(A1 , · · · , An )) = nj=1 ωj Φj (Aj ), in inequality (2.9), then ! n n X X βf I + ωj Φj f (Aj ) ≥ f ωj Φj (Aj ) , (2.13) j=1

j=1

where βf = maxm≤t≤M {f (t) − µf t − νf }; see also [7, Corollary 2.16]. We can generalize the operator Bellman inequality in Corollary 2.2 of [12] as follows Φ IH −

n X

ωj Aj

j=1

!!p

≥Φ

n X

ωj (IH − Aj )p

j=1

!

,

(2.14)

where Φ ∈ PN [(H ), (K )], Aj are contractions, 0 < p < 1 and ωj are real positive P numbers such that nj=1 ωj = 1.

In [12, Theorem 2.5], the authors presented an equivalent form of Bellman in-

equality. With similar proof, the following theorem holds Theorem 2.7. The following equivalent statements hold: (i) If m, n are positive integers, 0 < p < 1, ω1 , · · · , ωn ∈ R+ are any finite numP ber of positive real numbers such that nj=1 ωj = 1 and aij (j = 1, · · · n, i = P p 1, · · · , m) are positive real numbers such that m i=1 aij ≤ 1 for all j = 1, · · · n, then

n X

ωj

1−

j=1

m X i=1

1 p

aij

!p

≤

1−

m X n X i=1

j=1

ωj aij

1p

!p

.

(2.15)

(ii) (Classical Bellman Inequality) If n is a positive integer, 0 < p < 1 and Mi , aij (j = 1, · · · n, i = 1, · · · , m) are nonnegative real numbers such that Pm 1/p 1/p for all j = 1, · · · n, then i=1 aij ≤ Mj ! p1 p !p  n ! p1 m n n m 1 1 X X X X X − aij  . (2.16) Mjp − aijp ≤ Mj j=1

i=1

j=1

i=1

j=1

Now, we state reverse of (2.14) by following result.

Corollary 2.8 (A second type reverse operator Bellman inequality). Let Aj , Φj , ωj , j = 1, · · · , n be as above, Aj be contractions such that 0 ≤ mIH ≤ Aj ≤ MIH < IH


9

and 0 < p < 1. Then δIK +

n X

ωj Φj ((IH − Aj )p ) ≥

j=1

where δ = (1 − p) In particular,

ωj Φj (IH − Aj )

n X

p p−1

+

ωj (IH − Aj )p

j=1

!p

,

(2.17)

!!p

(2.18)

j=1

1 (1−m)p −(1−M )p p M −m

δIK + Φ

n X

(1−M )(1−m)p −(1−m)(1−M )p . M −m

!

≥

Φ

n X

(IH − Aj )

j=1

.

Proof. Note that the function g(t) = tr is operator concave on (0, ∞) when 0 ≤ r ≤ 1 and so is the function f (t) = (1 − t)p on (0, 1) when 0 ≤ p ≤ 1. It follows from the linearity and the normality of Φj that !p !p n n X X ωj Φj (IH − Aj ) = ωj IK − Φj (Aj ) j=1

j=1

=

IK −

n X

ωj Φj (Aj )

j=1

=f

n X

ωj Φj (Aj )

j=1

≤

n X

!p

!

ωj Φj (f (Aj )) + βf IK

(by (2.13))

j=1

=

n X

ωj Φj ((IH − Aj )p ) + βf IK .

j=1

Since f (t) = (1 − t)p is a differentiable function on (0, 1), the function h(t) := p )p )p −(1−m)p (1 − t)p − (1−MM t − M (1−m)M−m(1−M (m ≤ t ≤ M) attained its maximum −m −m 1 p )p p−1 value in t0 = 1 − p1 (1−m)M−(1−M which is equal to −m δ = max h(t) = (1 − p) m≤t≤M

1 (1 − m)p − (1 − M )p p M −m

p p−1

+

(1 − M )(1 − m)p − (1 − m)(1 − M )p . M −m

For inequality (2.18), it is enough to put Φj = Φ (j = 1, · · · , n) and use the concavity of f and linearity of Φ.

Corollary 2.9. If m, n are positive integers, 0 < p < 1, ω1 , · · · , ωn ∈ R+ are Pn any finite number of positive real numbers such that j=1 ωj = 1 and aij (j =

10


Pm

1, · · · n, i = 1, · · · , m) are positive real numbers such that 1 ≥

i=1

1, · · · n), then (1 − p)p

p 1−p

+

n X

ωj

1−

j=1

m X

1 p

aij

i=1

!p

≥

1−

m X n X

1 p

ωj aij

i=1 j=1

!p

1/p

aij

.

(j =

(2.19)

Proof. Let m, n be positive integers, 0 < p < 1, 0 ≤ ωj ≤ 1 and aij (1 ≤ j ≤ P 1/p n, 1 ≤ i ≤ m) are positive real numbers such that 1 ≥ m i=1 aij (j = 1, · · · n). Set # " P 1/p m 0 i=1 aij ∈ M2 (C) (j = 1, · · · , n). Then Aj = 0 1 n X

ωj (I2 − Aj )p =

n X

1 0

#

1

" P m

p i=1 aij 0

#!p

− 0 1 0 1 j=1   p 1 P n p X 1− m a 0 i=1 ij  = ωj  j=1 0 0   Pm p1 p Pn 0 ωj 1 − i=1 aij . =  j=1 0 0

j=1

ωj

"

and n X

ωj (I2 − Aj )

j=1

!p

=

n X

ωj

j=1

=

" P n

j=1

"

1 0 0 1

ωj (1 −

#

Pm

−

i=1

" P m

1

p i=1 aij 0

1

aijp ) 0

0 #p

1

#!!p

0 0   p 1 Pn Pm p 1 − j=1 i=1 ωj aij 0 . = 0 0 It follows from (2.18) with the identity map Φ that p     1 Pn Pm Pm p1 p Pn p 0 0 1 − j=1 i=1 ωj aij ωj 1 − i=1 aij ≥ , δI2 +  j=1 0 0 0 0 p

where δ = (1 − p)p 1−p by taking m = 0 and M = 1 in (2.17), which gives (2.19).


11

Corollary 2.10. Let Φ ∈ PN [B(H ), B(K )], 0 < mIH ≤ Aj ≤ MIH be positive P operators and 0 ≤ ωj ≤ 1 (j = 1, · · · , n) such that nj=1 ωj = 1. Then " # ! ! 1 n n X X 1 M m M −m log L(m, M) + Φ ωj log Aj ≥ log ωj Φ(Aj ) e mM j=1 j=1 where L(a, b) =

 

b−a log b−log a

; a 6= b

a

a and b.

is the Logarithmic mean of positive real numbers

;a = b

Proof. Put f (t) = log t and Φj = Φ in (2.13).

3. Some refinements of the Bellman operator inequality In this section, we present some refinements of the operator Bellman inequality by using some ideas of [4]. First we need the following Lemmas. P Lemma 3.1. Let A, B, Aj , Bj , (1 ≤ j ≤ n) be positive operators such that nj=1 Aj ≤ P A, nj=1 Bj ≤ B and let σf be an operator mean with the representing function f .

Then

A−

n X

Aj

j=1

!

σf

B−

n X

Bj

j=1

!

≤ (Aσf B) −

n X

(Aj σf Bj ) .

Proof. The subadditivity of operator mean says that [7, Theorem 5.7] ! ! n+1 n+1 n+1 X X X (Xj σf Yj ) ≤ Xj σf Yj , j=1

(3.1)

j=1

j=1

(3.2)

j=1

where Xj , Yj , (1 ≤ j ≤ n + 1) are positive operators. If we put Xj = Aj , Yj = P P Bj (1 ≤ j ≤ n), Xn+1 = A − nj=1 Aj and Yn+1 = A − nj=1 Bj in inequality (3.2), then we reach

n X

(Aj σf Bj ) +

j=1

A−

n X

Aj

j=1

!

σf

B−

n X

Bj

j=1

!

≤ Aσf B.

Therefore A−

n X j=1

Aj

!

σf

B−

n X j=1

Bj

!

≤ (Aσf B) −

n X

(Aj σf Bj ) .

j=1

12


Lemma 3.2. [11, Lemma 2.1] Let A, B ∈ B(H ) be positive operators such that A is contraction, h is a nonnegative operator monotone function on [0, +∞) and σf be an operator mean with the representing function f . Then

Aσhof B ≤ h(Aσf B).

In the next theorem, we show a refinement of (1.1). P Theorem 3.3. Let Aj , Bj , (1 ≤ j ≤ n) be positive operators such that nj=1 Aj ≤ Pn IH , j=1 Bj ≤ IH , σf be an operator mean with the representing function f and p ∈ [0, 1]. Then 

IH −

n X j=1





Aj  σf p IH −

n X j=1





Bj  ≤ IH − 

≤ IH −

k X j=1

n X j=1





Aj  σf IH − p

k X j=1



Bj  −

n X

j=k+1

p

(Aj σf Bj )

(Aj σf Bj ) ,

in which k = 1, 2, · · · , n − 1. Proof. For k = 1, 2, · · · , n − 1 we have



IH −

n X j=1





Aj σf IH −

n X j=1



Bj 

   n k n k X X X X Bj  Bj − Aj  σf  IH − Aj − =  IH − 



j=1

≤ IH −

k X j=1



n X j=1



Aj  σf IH −

≤ (IH σf IH ) − = IH −

j=1

j=k+1



k X j=1

k X j=1



Bj  −

(Aj σf Bj ) −

(Aj σf Bj ) ,



n X

n X

j=k+1

(Aj σf Bj ) (by (3.1))

j=k+1

(Aj σf Bj )

(by (3.1))

j=k+1

(3.3)


13

whence for the spacial case g(t) = tp (p ∈ [0, 1]) we have

IH −

n X j=1

!

Aj σf p

IH −

n X

Bj

j=1

≤

IH −

n X

!

Aj

j=1

≤

IH −

k X

Aj

j=1

=

IH −

n X

! !

σf

IH −

n X

Bj

!!p

Bj

!

j=1

σf

IH −

k X j=1

(Aj σf Bj )

j=1

!p

−

(by Lemma 3.2) n X

(Aj σf Bj )

j=k+1

!p

.

Using the same idea as in the proof of Theorem 3.3 we improve inequality (1.1) in the next theorem.

P Theorem 3.4. Let Aj , Bj , (1 ≤ j ≤ n) be positive operators such that nj=1 Aj ≤ P IH , nj=1 Bj ≤ IH , σf be an operator mean with the representing function f and p ∈ [0, 1]. Then 

IH −

n X j=1





Aj σf p IH − 

n X j=1

≤ IH − 

≤ IH −

Bj 

n X j=1

n X j=1

 



tj Aj  σf IH − p

(Aj σf Bj ) ,

where tj ∈ [0, 1] (j = 1, · · · , n).

n X j=1



tj B j  −

n X j=1

p

(1 − tj ) (Aj σf Bj )

14


P P Proof. Let tj ∈ [0, 1] (j = 1, · · · , n). It follows from IH − nj=1 tj Aj ≥ nj=1 (1 − P P tj )Aj , IH − nj=1 tj Bj ≥ nj=1(1 − tj )Bj and inequality (3.1) that ! ! n n X X IH − Aj σf IH − Bj j=1

=

≤

=

IH −

j=1

n X

tj Aj −

j=1

n X

(1 − tj )Aj

j=1

!

σf

IH −

n X

tj Bj −

j=1

n X

(1 − tj )Bj

j=1

n n n X X X IH − tj Aj σf IH − tj Bj − ((1 − tj )Aj σf (1 − tj )Bj )

IH −

j=1

j=1

n X

n X

j=1

tj Aj σf IH −

j=1

tj Bj

j=1

n X − (1 − tj ) (Aj σf Bj ) j=1

!

!

(by (3.1))

!

(by property (iii) of operator means) ≤

(IH σf IH ) −

n X

(tj Aj σf tj Bj ) −

j=1

=

IH −

n X j=1

=

IH

tj (Aj σf Bj ) −

n X

!

(1 − tj ) (Aj σf Bj )

j=1

n X

(1 − tj ) (Aj σf Bj )

j=1

(by (3.1))

!

(by property (iii) of operator means) ! n X − (Aj σf Bj ) .

(3.4)

j=1

Using Lemma 3.2, the operator monotone function g(t) = tp (p ∈ [0, 1]) and inequality (3.4) we get the desired result.

Acknowledgement. The authors would like to sincerely thank the anonymous referee for some useful comments and suggestions. The first author would like to thank the Tusi Mathematical Research Group (TMRG).

References 1. J. Acz´ el, Some general methods in the theory of functional equations in one variable, New applications of functional equations. (Russian) Uspehi Mat. Nauk (N.S.) 11, 3(69) (1956), 3–68. 2. E.F. Beckenbach, R. Bellman, Inequalities, Springer, 1971. 3. R. Bellman, On an inequality concerning an indefinite form, Amer. Math. Monthly 63 (1956), 101–109.


15

4. J.L. Daz-Barrero, M. Grau-S´ anchez and P. G. Popescu, Refinements of Aczél, Popoviciu and Bellman’s inequalities, Comput. Math. Appl. (2) (2008), 2356–2359. 5. M. Fujii, Y.O. Kim and R. Nakamoto, A characterization of convex functions and its application to operator monotone functions, Banach J. Math. Anal. 8(2) (2014), 118–123. 6. M. Fujii, J. Mićić Hot, J. Peˇcarić and Y. Seo, Recent developments of Mond–Peˇcarić method in operator inequalities. Inequalities for bounded selfadjoint operators on a Hilbert space. II., Monographs in Inequalities 4. Zagreb: Element, 2012. 7. T. Furuta, J. Mićić Hot, J. Peˇcarić and Y. Seo, Mond–Peˇcarić method in operator inequalities, Zagreb, 2005. 8. R. Kaur, M. Singh and J.S. Aujla, Generalized matrix version of reverse H¨ older inequality, Linear Algebra Appl. 434(3) (2011), 636–640. 9. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980), 205-224. 10. A. Matsumoto and M. Tominaga, Mond–Peˇcarić method for a mean-like transformation of operator functions, Sci. Math. Jpn. 61(2) (2005), 243-247. 11. F. Mirzapour, M.S. Moslehian and A. Morassaei, More on operator Bellman inequality, Quaest. Math. 37(1) (2014), 9–17. 12. M. Morassaei, F. Mirzapour and M.S. Moslehian, Bellman inequality for Hilbert space operators, Linear Algebra Appl. 438 (2013), 3776–3780. 13. M.S. Moslehian, Operator Aczél inequality, Linear Algebra Appl. 434 (2011), 1981–1987. 14. V.I. Paulsen, Completely bounded maps and dilation, Pitman Research Notes in Mathematics Series, 146, John Wiley & Sons, Inc., New York, 1986. 15. T. Popoviciu, On an inequality, Gaz. Mat. Fiz. Ser. A, 11(64) (1959), 451–461. 16. S. Wu and L. Debnath, A new generalization of Aczèls inequality and its applications to an improvement of Bellmans inequality, Appl. Math. Lett. 21(6) (2008), 588–593. 1

Department of Mathematics, Faculty of Mathematics, University of Sistan and

Baluchestan, Zahedan, Iran. E-mail address: [email protected]; [email protected] 2

Department of Mathematics, Faculty of Sciences, University of Zanjan, Uni-

versity Blvd., Zanjan 45371-38791, Iran E-mail address: [email protected], [email protected]