Quasilinearity of some functionals associated to a weakened davis-choi-jensen’s inequality for positive maps

V. 10,No 2,2020,July ISSN 2218-6816

Quasilinearity of Some Functionals Associated to a Weakened Davis-Choi-Jensen’s Inequality for

Positive Maps

S.S. Dragomir

Abstract. In this paper we establish some quasilinearity properties of some functionals associated to a weakened Davis-Choi-Jensen’s inequality for positive maps and convex (concave) functions. Applications for power function are also provided.

Key Words and Phrases: Jensen’s inequality, convex functions, positive maps, super- additivity, subadditivity.

2010 Mathematics Subject Classifications: 47A63, 47A30, 15A60

1. Introduction

Let H be a complex Hilbert space and B(H) be the Banach algebra of bounded linear operators acting on H. We denote by B_h(H) the semi-space of all selfadjoint operators in B(H).We denote by B⁺(H) the convex cone of all positive operators onH and by B⁺⁺(H) the convex cone of all positive definite operators onH.

Let H, K be complex Hilbert spaces. Following [2] (see also [15, p. 18]) we can introduce the following definition:

Definition 1. A map Φ :B(H) → B(K) is linear if it is additive and homoge- neous, namely

Φ (λA+µB) =λΦ (A) +µΦ (B)

for any λ, µ ∈ C and A, B ∈ B(H). The linear map Φ : B(H) → B(K) is positive if it preserves the operator order, i.e. if A ∈ B⁺(H), then Φ (A) ∈ B⁺(K). We write Φ ∈ P[B(H),B(K)]. The linear map Φ : B(H) → B(K) is normalised if it preserves the identity operator, i.e. Φ (1_H) = 1_K. We write Φ∈P_N[B(H),B(K)].

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We observe that a positive linear map Φ preserves the order relation, namely A≤B implies Φ (A)≤Φ (B)

and preserves the adjoint operation Φ (A^∗) = Φ (A)^∗.If Φ ∈P_N[B(H),B(K)]

and α1H ≤A≤β1H,thenα1K ≤Φ (A)≤β1K.

If the map Ψ :B(H)→ B(K) is linear, positive and Ψ (1_H)∈ B⁺⁺(K), then by putting Φ = Ψ^−1/2(1_H) ΨΨ^−1/2(1_H) we get that Φ ∈ P_N[B(H),B(K)], namely it is also normalised.

A real valued continuous function f on an interval I is said to be operator convex (concave) onI if

f((1−λ)A+λB)≤(≥) (1−λ)f(A) +λf(B)

for allλ∈[0,1] and for every selfadjoint operators A, B ∈ B(H) whose spectra are contained inI.

The following Jensen’s type result is well known [2]:

Theorem 1 (Davis-Choi-Jensen’s Inequality). Let f : I → R be an operator convex function on the interval I and Φ ∈ P_N[B(H),B(K)]. Then for any selfadjoint operatorA whose spectrum is contained in I we have

f(Φ (A))≤Φ (f(A)). (1)

We observe that if Ψ ∈ P[B(H),B(K)] with Ψ (1H) ∈ B⁺⁺(K), then by taking Φ = Ψ^−1/2(1_H) ΨΨ^−1/2(1_H) in (1) we get

f

Ψ^−1/2(1_H) Ψ (A) Ψ^−1/2(1_H)

≤Ψ^−1/2(1_H) Ψ (f(A)) Ψ^−1/2(1_H). If we multiply both sides of this inequality by Ψ^1/2(1_H) we get the following Davis-Choi-Jensen’s inequality for general positive linear maps:

Ψ^1/2(1_H)f

Ψ^−1/2(1_H) Ψ (A) Ψ^−1/2(1_H)

Ψ^1/2(1_H)≤Ψ (f(A)). (2) In the recent paper [9] we established the followingweakened version of Davis- Choi-Jensen’s inequality that holds for the larger class of convex functions:

Theorem 2. Let f : I → R be a convex function on the interval I and Φ : B(H)→ B(K) a normalised positive linear map. Then for any selfadjoint oper- ator A whose spectrum Sp(A) is contained inI we have

f(hΦ (A)y, yi)≤ hΦ (f(A))y, yi (3) for anyy ∈K, kyk= 1.

For the sake of completeness, we give here a simple proof as follows.

Let m < M and Sp(A) ⊆ [m, M] ⊂ I. Then m1_H ≤ A ≤ M1_H and since Φ∈P_N[B(H),B(K)] we havem1K ≤Φ (A)≤M1Kshowing thathΦ (A)y, yi ∈ [m, M] for anyy∈K, kyk= 1.

By the gradient inequality for the convex functionfwe have fora=hΦ (A)y, yi

∈[m, M]

f(t)≥f(hΦ (A)y, yi) + (t− hΦ (A)y, yi)f₊⁰ (hΦ (A)y, yi) for any t∈I,where f₊⁰ is the lateral derivative off on I.

Using the continuous functional calculus for the operator A we have for a fixed y∈K withkyk= 1

f(A)≥f(hΦ (A)y, yi) 1_H +f₊⁰ (hΦ (A)y, yi) (A− hΦ (A)y, yi1H). (4) Since Φ∈P_N[B(H),B(K)],by taking the functional Φ in the inequality (4) we get

Φ (f(A))≥f(hΦ (A)y, yi) 1_K+f₊⁰ (hΦ (A)y, yi) (Φ (A)− hΦ (A)y, yi1K) (5) for any y∈K withkyk= 1.

This inequality is of interest in itself.

Taking the inner product in (5) we have for any y∈K withkyk= 1 hΦ (f(A))y, yi

≥f(hΦ (A)y, yi)kyk²+f₊⁰ (hΦ (A)y, yi)

hΦ (A)y, yi − hΦ (A)y, yi kyk²

=f(hΦ (A)y, yi)

and the inequality (3) is proved.

If the normality condition is dropped, then we have:

Corollary 1. Let f : I → R be a convex function on the interval I and Ψ ∈ P[B(H),B(K)] with Ψ (1_H) ∈ B⁺⁺(K). Then for any selfadjoint operator A whose spectrum Sp(A) is contained inI we have

f

hΨ (A)v, vi hΨ (1_H)v, vi

≤ hΨ (f(A))v, vi

hΨ (1_H)v, vi (6) for anyv ∈K withv 6= 0.

For Jensen’s type operator inequalities see [1], [3]-[14] and the references therein.

We define by P_I[B(H),B(K)] the convex cone of all linear, positive maps Ψ with Ψ (1H)∈ B⁺⁺(K),namely Ψ (1H) is a positive invertible operator in K and define the functional4_f,A,v :P_I[B(H),B(K)]→ B(K) by

4_f,A,v(Ψ) =hΨ (1_H)v, vif

hΨ (A)v, vi hΨ (1_H)v, vi

,

wheref :I →Ris a convex (concave) function on the intervalI,Ais a selfadjoint operator whose spectrum is contained inI and v∈K, v6= 0.

In this paper we establish some quasilinearity properties of some function- als associated to the weakened Davis-Choi-Jensen’s inequality (6) for positive maps and convex (concave) functions. Applications for power function are also provided.

2. The main results The following result holds:

Theorem 3. Let f :I →R be a convex (concave) function on the interval I, A a selfadjoint operator whose spectrum is contained in I and v∈K, v 6= 0. If Ψ1, Ψ₂∈P_I[B(H),B(K)] andλ∈[0,1], then

4_f,A,v((1−λ) Ψ₁+λΨ₂)≤(≥) (1−λ)4_f,A,v(Ψ₁) +λ4_f,A,v(Ψ₂), (7) namely4_f,A,v is convex (concave) on P_I[B(H),B(K)].

In particular, we have

4_f,A,v(Ψ₁+ Ψ₂)≤(≥)4_f,A,v(Ψ₁) +4_f,A,v(Ψ₂), (8) namely4_f,A,v is subadditive (superadditive) on P_I[B(H),B(K)].

Proof. Assume that f : I → R is a convex function on the interval I and v∈K, v6= 0.

Let Ψ₁,Ψ₂ ∈P_I[B(H),B(K)] and λ∈[0,1]. Then

4_f,A,v((1−λ) Ψ₁+λΨ₂) (9)

=h((1−λ) Ψ1+λΨ2) (1_H)v, vif

h((1−λ) Ψ1+λΨ2) (A)v, vi h((1−λ) Ψ₁+λΨ₂) (1_H)v, vi

= [(1−λ)hΨ₁(1H)v, vi+λhΨ₂(1H)v, vi]

×f

(1−λ)hΨ₁(A)v, vi+λhΨ₂(A)v, vi (1−λ)hΨ₁(1H)v, vi+λhΨ₂(1H)v, vi

.

Using the convexity of f we have f

(1−λ)hΨ₁(A)v, vi+λhΨ₂(A)v, vi (1−λ)hΨ₁(1_H)v, vi+λhΨ₂(1_H)v, vi

(10)

=f





(1−λ)hΨ₁(1_H)v, vi_hΨ^hΨ1(A)v,vi

1(1H)v,vi +λhΨ₂(1_H)v, vi_hΨ^hΨ2(A)v,vi

2(1H)v,vi

(1−λ)hΨ₁(1H)v, vi+λhΨ₂(1H)v, vi





≤

(1−λ)hΨ₁(1_H)v, vif_hΨ

1(A)v,vi hΨ1(1H)v,vi

+λhΨ₂(1_H)v, vif_hΨ

2(A)v,vi hΨ2(1H)v,vi

(1−λ)hΨ₁(1H)v, vi+λhΨ₂(1H)v, vi

and by multiplying (10) by (1−λ)hΨ₁(1_H)v, vi+λhΨ₂(1_H)v, vi > 0 and by using (9), we get

4_f,A,v((1−λ) Ψ1+λΨ2)

≤(1−λ)hΨ₁(1H)v, vif

hΨ₁(A)v, vi hΨ₁(1H)v, vi

+λhΨ₂(1H)v, vif

hΨ₂(A)v, vi hΨ₂(1H)v, vi

= (1−λ)4_f,A,v(Ψ₁) +λ4_f,A,v(Ψ₂), which proves the convexity of4_f,A,v.

We have by (7)

4_f,A,v(Ψ1+ Ψ2) =4_f,A,v

2Ψ1+ 2Ψ2

2

≤ 4_f,A,v(2Ψ₁) +4_f,A,v(2Ψ₂) 2

= 24_f,A,v(Ψ₁) + 24_f,A,v(Ψ₂)

2 =4_f,A,v(Ψ1) +4_f,A,v(Ψ2) for any Ψ1,Ψ2∈P_I[B(H),B(K)],which proves (8). J

For Ψ1, Ψ2 ∈ P_I[B(H),B(K)] we denote Ψ2 _I Ψ1 if Ψ2 −Ψ1 ∈ P_I[B (H),B(K)],see also [10]. This means that Ψ2−Ψ1 is a linear positive functional and Ψ₂(1_H)−Ψ₁(1_H)∈ B⁺⁺(K).

We have:

Corollary 2. Let f : I → [0,∞) be a concave function on the interval I, A a selfadjoint operator whose spectrum is contained in I and v∈K, v 6= 0.

(i) If Ψ1,Ψ2∈P_I[B(H),B(K)] withΨ2 _I Ψ1, then

4_f,A,v(Ψ₂)≥ 4_f,A,v(Ψ₁), (11)

namely4_f,A,v is operator monotonic in the order”_I ” of P_I[B(H),B(K)]. (ii) If Ψ,Υ ∈P_I[B(H),B(K)], t, T > 0 with T > t and TΥ_I Ψ_I tΥ, then

T4_f,A,v(Υ)≥ 4_f,A,v(Ψ)≥t4_f,A,v(Υ). (12)

Proof. (i) Let Ψ₁, Ψ₂ ∈ P_I[B(H),B(K)] with Ψ_{2 I} Ψ₁. Then by (8) we have

4_f,A,v(Ψ₂) =4_f,A,v(Ψ₁+ Ψ₂−Ψ₁)≥ 4_f,A,v(Ψ₁) +4_f,A,v(Ψ₂−Ψ₁) implying that

4_f,A,v(Ψ₂)− 4_f,A,v(Ψ₁)≥ 4_f,A,v(Ψ₂−Ψ₁).

Since f is positive and Ψ₂−Ψ₁ ∈ P_I[B(H),B(K)] with Ψ₂(1_H)−Ψ₁(1_H) ∈ B⁺⁺(K), it follows that 4_f,A,v(Ψ2−Ψ1)≥0 and the inequality (11) is proved.

(ii) The proof follows by (11) on taking first Ψ₂ = TΥ, Ψ₁ = Ψ and then Ψ₂= Ψ,Ψ₁ =tΥ and by the positive homogeneity of4_f,A,v.J

We consider now the functional 4_f,A,v :P_I[B(H),B(K)] → B(K) defined by

f,A,v(Ψ) :=hΨ (f(A))v, vi − 4_f,A,v(Ψ) (13)

=hΨ (f(A))v, vi − hΨ (1_H)v, vif

hΨ (A)v, vi hΨ (1_H)v, vi

,

wheref :I →Ris a convex (concave) function on the intervalI,Ais a selfadjoint operator whose spectrum is contained inI and v∈K, v6= 0.

We can state the following result:

Theorem 4. Let f : I → R be a convex (concave) function on the interval I and Aa selfadjoint operator whose spectrum is contained in I andv∈K, v6= 0.

Then the functional f,A,v is positive (negative) on P_I[B(H),B(K)], and it is positive homogeneous and concave (convex) on P_I[B(H),B(K)].f,A,v is also superadditive (subadditive) onP_I[B(H),B(K)].

Proof. We consider only the convex case. The positivity off,A,vonP_I[B(H), B(K)] is equivalent to the inequality for general positive linear maps (6). The positive homogeneity follows by the same property of 4_f,A,v and the definition of 4_f,A,v.

If Ψ1,Ψ2∈P_I[B(H),B(K)], λ∈[0,1] andv∈K, v6= 0,then by Theorem 3 we have

f,A,v((1−λ) Ψ1+λΨ2)

=h((1−λ) Ψ₁+λΨ₂) (f(A))v, vi − 4_f,A,v((1−λ) Ψ₁+λΨ₂)

≥(1−λ)hΨ₁(f(A))v, vi+λh(Ψ₂f(A))v, vi

−(1−λ)4_f,A,v(Ψ₁)−λ4_f,A,v(Ψ₂)

= (1−λ) [hΨ₁(f(A))v, vi − 4_f,A,v(Ψ₁)]

+λ[h(Ψ₂f(A))v, vi − 4_f,A,v(Ψ₂)]

= (1−λ)f,A,v(Ψ₁) +λf,A,v(Ψ₂) that proves the operator concavity of f,A,v.

The operator superadditivity follows in a similar way and we omit the details.

J

Corollary 3. Let f : I → R be a convex function on the interval I, A a self- adjoint operator whose spectrum is contained in I and v ∈ K, v 6= 0. If Ψ, Υ∈P_I[B(H),B(K)], t, T >0 withT > t andTΥ_I Ψ_I tΥ, then

Tf,A,v(Υ)≥f,A,v(Ψ)≥tf,A,v(Υ) (14) or, equivalently,

T(hΥ (f(A))v, vi − 4_f,A,v(Υ))≥ hΨ (f(A))v, vi − 4_f,A,v(Ψ) (15)

≥t(hΥ (f(A))v, vi − 4_f,A,v(Υ))≥0.

Now, assume that Ais a selfadjoint operator whose spectrum is contained in [m, M] for some real constants M > m. If f is convex, then for any t∈ [m, M]

we have

f(t)≤ (M −t)f(m) + (t−m)f(M)

M−m . (16)

IfAis a selfadjoint operator whose spectrum is contained in [m, M],thenm1H ≤ A ≤ M1_H and by taking the map Ψ we get mΨ (1_H) ≤ Ψ (A) ≤ MΨ (1_H) for Ψ∈P_I[B(H),B(K)].This is equivalent to

m≤ hΨ (A)v, vi hΨ (1_H)v, vi ≤M for any v∈K, v 6= 0.

If we take t= _hΨ(1^hΨ(A)v,vi

H)v,vi, v∈K, v 6= 0 in (16), then we get

f

hΨ (A)v, vi hΨ (1_H)v, vi

≤

(M−_hΨ(1^hΨ(A)v,vi

H)v,vi)f(m) + _hΨ(A)v,vi

hΨ(1H)v,vi−m f(M) M−m

that is equivalent to

4_f,A,v(Ψ)≤♦f,A,v(Ψ) where

♦f,A,v(Ψ) := h(MΨ (1H)−Ψ (A))v, vif(m) +h(Ψ (A)−mΨ (1H))v, vif(M) M−m

for Ψ ∈ P_I[B(H),B(K)], is a trapezoidal type functional. We observe that

♦f,A,v isadditive and positive homogeneous onP_I[B(H),B(K)]. We define the functional †_f,A,v :P_I[B(H),B(K)]→ B(K) by

†_f,A,v(Ψ) :=♦f,A,v(Ψ)− 4_f,A,v(Ψ)

= h(MΨ (1_H)−Ψ (A))v, vif(m) +h(Ψ (A)−mΨ (1_H))v, vif(M) M−m

− hΨ (1_H)v, vif

hΨ (A)v, vi hΨ (1_H)v, vi

.

We observe that if f is convex (concave) on [m, M] and m1H ≤A≤M1H,then

†_f,A,v(Ψ)≥(≤) 0 for any Ψ∈P_I[B(H),B(K)]. (17) Theorem 5. Let f : I → R be a convex (concave) function on the interval I and A a selfadjoint operator whose spectrum is contained in [m, M] and v ∈K, v6= 0.Then the functional†_f,A,v is positive (negative) onP_I[B(H),B(K)],and it is positive homogeneous and concave (convex) on P_I[B(H),B(K)]. †_f,A,v is also superadditive (subadditive) on P_I[B(H),B(K)].

The proof is similar to the one of Theorem 4 and we omit the details.

Corollary 4. Let f : I → R be a convex function on the interval I, A a self- adjoint operator whose spectrum is contained in I and v ∈ K, v 6= 0. If Ψ, Υ∈P_I[B(H),B(K)], t, T >0 withT > t andTΥ_I Ψ_I tΥ, then

T†_f,A,v (Υ)≥ †_f,A,v(Ψ)≥t†_f,A,v(Υ) (18) or, equivalently,

T

h(MΥ (1_H)−Υ (A))v, vif(m) +h(Υ (A)−mΥ (1_H))v, vif(M)

M−m (19)

− hΥ (1_H)v, vif

hΥ (A)v, vi hΥ (1_H)v, vi

≥ h(MΨ (1_H)−Ψ (A))v, vif(m) +h(Ψ (A)−mΨ (1_H))v, vif(M) M−m

− hΨ (1_H)v, vif

hΨ (A)v, vi hΨ (1_H)v, vi

≥t

h(MΥ (1_H)−Υ (A))v, vif(m) +h(Υ (A)−mΥ (1_H))v, vif(M) M−m

− hΥ (1_H)v, vif

hΥ (A)v, vi hΥ (1_H)v, vi

≥0.

3. Some examples

Let A_i be selfadjoint operators onH with Sp(A_i)⊂I, i∈ {1, ..., n} andp= (p₁, ..., p_n) an n-tuple of nonnegative weights withP_n:= Pn

i=1p_i >0.We write p∈Rⁿ++.Consider also the n-tuple of normalised positive maps Φ = (φ1, ..., φn) withφ_i ∈P_N[B(H),B(H)] fori∈ {1, ..., n}.

If we put

A˜:=







A₁ · · · 0 ... . .. ... 0 · · · A_n





,

then we haveSp A˜

⊂I.We can define the positive map Ψ_p,Φ:B(H)⊕...⊕ B(H)→ B(H) by

Ψ_p,Φ(A₁⊕...⊕A_n) =

n

X

i=1

p_iφ_i(A_i).

Using the functional calculus for continuous functionsf onI we have Ψp,Φ

f

A˜

=

n

X

i=1

piφi(f(Ai)) and f

Ψp,Φ

A˜

=f

n

X

i=1

piφi(Ai)

! .

Since

Ψ_p,Φ(1_H ⊕...⊕1_H) =

n

X

i=1

p_iφ_i(1_H) =P_n1_H and P_n>0 it follows that Ψ_p,Φ ∈P_I[B(H)⊕...⊕ B(H),B(H)].

If p, q∈Rⁿ₊₊with p≥q,namely pi ≥qi fori∈ {1, ..., n}and Pn> Qn, then Ψ_{p,Φ I}Ψ_q,Φ.

Assume also thatr= mini∈{1,...,n}

npi

qi

o

,R= maxi∈{1,...,n}

npi

qi

o

andr < _Q^Pn

n < R.

Then

Ψ_p,Φ A˜

−rΨ_q,Φ A˜

=

n

X

i=1

(p_i−rq_i)φ_i(A_i)≥0 for ˜A≥˜0,

Ψ_p,Φ ˜1_H

−rΨ_q,Φ ˜1_H

=

n

X

i=1

(p_i−rq_i)φ_i(1_H) = (P_n−rQ_n) 1_H

and

RΨ_q,Φ ˜1_H

−Ψ_p,Φ ˜1_H

=

n

X

i=1

(Rq_i−p_i)φ_i(1_H) = (RQ_n−P_n) 1_H showing that

RΨq,Φ_I Ψp,Φ _IrΨq,Φ. (20) Now, observe that for v∈H,kvk= 1 we have

4_f,A,v˜ (Ψ_p,Φ) =P_nf hPn

i=1piφi(Ai)v, vi P_n

, wherep∈Rⁿ++.

Letf :I →Rbe a convex (concave) function on the intervalI,Aa selfadjoint operator whose spectrum is contained in I and v ∈ H, kvk = 1. If p, q ∈ Rⁿ₊₊, then we have by Theorem 3 that

4_f,A,v˜ Ψ(1−λ)p+λq,Φ

≤(≥) (1−λ)4_f,A,v˜ (Ψ_p,Φ) +λ4_f,A,v˜ (Ψ_q,Φ) (21) for any λ∈[0,1] and, in particular,

4_f,A,v˜ (Ψp+q,Φ)≤(≥)4_f,A,v˜ (Ψp,Φ) +4_f,A,v˜ (Ψq,Φ). (22) By using (12) forp, q∈Rⁿ++withr= mini∈{1,...,n}

npi

qi

o

,R= maxi∈{1,...,n}

npi

qi

o and r < _Q^Pn

n < R we have RQ_nf

hPn

i=1q_iφ_i(A_i)v, vi Qn

≥P_nf hPn

i=1p_iφ_i(A_i)v, vi Pn

(23)

≥rQnf hPn

i=1qiφi(Ai)v, vi Q_n

,

provided f :I → [0,∞) is a concave function on the interval I, A a selfadjoint operator whose spectrum is contained inI and v∈H,kvk= 1.

If we take f(t) = t^s, s ∈ (0,1) and assume that A_i ≥ 0, i ∈ {1, ..., n}, then by (23) we have the power inequality

R^1/sQ^1/s−1_n

* _n X

i=1

qiφi(Ai)v, v +

≥P_n^1/s−1

* _n X

i=1

piφi(Ai)v, v +

(24)

≥r^1/sQ^1/s−1_n

* _n X

i=1

qiφi(Ai)v, v +

,

forv∈H,kvk= 1.

By taking the supremum in this inequality over v ∈ H,kvk = 1, we get the norm inequality

R^1/sQ^1/s−1_n

n

X

i=1

q_iφ_i(A_i)

≥P_n^1/s−1

n

X

i=1

p_iφ_i(A_i)

(25)

≥r^1/sQ^1/s−1_n

n

X

i=1

q_iφ_i(A_i) .

We also have _f,A,v˜ (Ψ_p,Φ) :=

n

X

i=1

pihφ_i(f(Ai))v, vi −Pnf hPn

i=1piφi(Ai)v, vi P_n

, (26) wherep∈Rⁿ++.

By utilising (15) we can state that R

" _n X

i=1

qihφ_i(f(Ai))v, vi −Qnf hPn

i=1q_iφ_i(A_i)v, vi Qn

#

(27)

≥

n

X

i=1

p_ihφ_i(f(A_i))v, vi −P_nf hPn

i=1piφi(Ai)v, vi P_n

≥r

" _n X

i=1

q_ihφ_i(f(A_i))v, vi −Q_nf hPn

i=1qiφi(Ai)v, vi Q_n

#

forp, q ∈Rⁿ++withr= mini∈{1,...,n}

npi

qi

o

,R= maxi∈{1,...,n}

npi

qi

o

andr < _Q^Pn

n <

R.

If we take f(t) =|t|^α, t∈R with α ≥ 1, then for any selfadjoint operators A_i, i∈ {1, ..., n} we have

R

" _n X

i=1

q_ihφ_i(|A_i|^α)v, vi −Q^1−α_n

* _n X

i=1

q_iφ_i(A_i)v, v +

α#

(28)

≥

n

X

i=1

p_ihφ_i(|A_i|^α)v, vi −P_n^1−α

* _n X

i=1

p_iφ_i(A_i)v, v +

α

≥r

" _n X

i=1

qihφ_i(|A_i|^α)v, vi −Q^1−α_n

* _n X

i=1

qiφi(Ai)v, v +

α# .

Finally, since

†_f,A,v˜ (Ψ_p,Φ) = 1 M−m

"*

M P_n1_H −

n

X

i=1

p_iφ_i(A_i)

! v, v

+ f(m) +

* _n X

i=1

p_iφ_i(A_i)−mP_n1_H

! v, v

+ f(M)

#

−P_nf hPn

i=1p_iφ_i(A_i)v, vi Pn

,

by (19) we have R

( 1 M−m

"*

M Q_n1_H−

n

X

i=1

q_iφ_i(A_i)

! v, v

+ f(m) +

* _n X

i=1

qiφi(Ai)−mQn1_H

! v, v

+ f(M)

#

−Q_nf hPn

i=1qiφi(Ai)v, vi Q_n

≥ 1

M−m

"*

M P_n1_H −

n

X

i=1

p_iφ_i(A_i)

! v, v

+ f(m) +

* _n X

i=1

p_iφ_i(A_i)−mP_n1_H

! v, v

+ f(M)

#

−P_nf hPn

i=1p_iφ_i(A_i)v, vi Pn

≥r ( 1

M−m

"*

M Q_n1_H−

n

X

i=1

q_iφ_i(A_i)

! v, v

+ f(m) +

* _n X

i=1

q_iφ_i(A_i)−mQ_n1_H

! v, v

+ f(M)

#

−Q_nf hPn

i=1q_iφ_i(A_i)v, vi Qn

forp, q∈Rⁿ++ withr= mini∈{1,...,n}

npi

qi

o

,R= maxi∈{1,...,n}

npi

qi

o

and r < _Q^Pn

n <

R.

Several other inequalities may be obtained if one chooses the convex functions f(t) =−lnt, tlnt, t^β, wheret >0 andβ ∈(−∞,0)∪[1,∞) orf(t) = exp (γt), t, γ ∈R andγ 6= 0.The details are omitted.

References

[1] R.P. Agarwal, S.S. Dragomir, A survey of Jensen type inequalities for functions of selfadjoint operators in Hilbert spaces, Comput. Math. Appl., 59(12), 2010, 3785–3812.

[2] M.D. Choi,Positive linear maps on C*-algebras, Canad. J. Math.,24, 1972, 520–529.

[3] S.S. Dragomir, Some reverses of the Jensen inequality for functions of self- adjoint operators in Hilbert spaces, J. Inequal. Appl.,2010, Art. ID 496821, 2010, 15 pp.

[4] S.S. Dragomir,A survey of Jensen type inequalities for log-convex functions of selfadjoint operators in Hilbert spaces, Commun. Math. Anal., 10(1), 2011, 82–104.

[5] S.S. Dragomir,Operator Inequalities of the Jensen, ˇCebyˇsev and Gr¨uss Type, Springer Briefs in Mathematics. Springer, New York, 2012. xii+121 pp.

ISBN: 978-1-4614-1520-6.

[6] S.S. Dragomir,Jensen type weighted inequalities for functions of selfadjoint and unitary operators, Ital. J. Pure Appl. Math., 32, 2014, 247–264.

[7] S.S. Dragomir,Some inequalities of Jensen type for Arg-square convex func- tions of unitary operators in Hilbert spaces, Bull. Aust. Math. Soc., 90(1), 2014, 65–73.

[8] S.S. Dragomir, Some Jensen type inequalities for square-convex functions of selfadjoint operators in Hilbert spaces, Commun. Math. Anal., 14(1), 2013, 42–58.

[9] S.S. Dragomir, A weakened version of Davis-Choi-Jensen’s inequality for normalised positive maps, Proyecciones Journal of Mathematics, 36(1), 2017, 81-94.

[10] S.S. Dragomir, Operator quasilinearity of some functionals associated to Davis-Choi-Jensen’s inequality for positive maps, Bull. Austral. Math Soc., 95(2), 2017, 322–332.

[11] M. Kian, Operator Jensen inequality for superquadratic functions, Linear Algebra Appl., 456, 2014, 82–87.

[12] M. Kian, M.S. Moslehian, Refinements of the operator Jensen-Mercer in- equality, Electron. J. Linear Algebra,26, 2013, 742–753.

[13] S.M. Malamud, Operator inequalities that are inverse to Jensen’s inequal- ity, Mat. Zametki,69(4), 2001, 633–637 (in Russian); translation in Math.

Notes,69(3-4), 2001, 582–586.

[14] A. Matkovi´c, J. Peˇcari´c, On a variant of the Jensen-Mercer inequality for operators, J. Math. Inequal.,2(3), 2008, 299–307.

[15] J. Peˇcari´c, T. Furuta, J. Mi´ci´c Hot, Y. Seo,Mond-Peˇcari´c Method in Opera- tor Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space, Element, Zagreb, 2005.

S.S. Dragomir

Mathematics, College of Engineering & Science Victoria University, PO Box 14428

Melbourne City, MC 8001, Australia E-mail: sever.dragomir@vu.edu.au

Received 15 August 2019 Accepted 30 January 2020