# Inequalities of Convex Functions and Self-Adjoint Operators

Authors: Zlatko Pavić
Journal: Journal of Operators
 (2014)

---

## Abstract

The paper offers generalizations of the Jensen-Mercer inequality for self-adjoint operators and generally convex functions. The obtained results are applied to define the quasi-arithmetic operator means without using operator convexity. The version of the harmonic-geometric-arithmetic operator
mean inequality is derived as an example.


---

## Body

## 1. Introduction

Throughout the paper we will use a real intervalℐ with the nonempty interior and real segments [
a
,
b
] and (
a
,
b
) with a
<
b.We briefly summarize a development path of the operator form of Jensen’s inequality. Letℋ and 𝒦 be Hilbert spaces, let B
(
ℋ
) and B
(
𝒦
) be associated C
*-algebras of bounded linear operators, and let 1
H and 1
K be their identity operators.Combining the results from [1, 2], it follows that every operator convex function f
:
ℐ
→
ℝ satisfies the Schwarz inequality
(1)
f
(
Φ
(
A
)
)
≤
Φ
(
f
(
A
)
)
,
where Φ
:
B
(
ℋ
)
→
B
(
𝒦
) is a positive linear mapping such that Φ
(
1
H
)
=
1
K and A
∈
B
(
ℋ
) is a self-adjoint operator with the spectrum Sp
(
A
)
⊆
ℐ. The above inequality was extended in [3] to the inequality
(2)
f
(
∑
i
=
1
n
‍
Φ
i
(
A
i
)
)
≤
∑
i
=
1
n
‍
Φ
i
(
f
(
A
i
)
)
,
where Φ
i
:
B
(
ℋ
)
→
B
(
𝒦
) are positive linear mappings such that ∑
i
=
1
n
‍
Φ
i
(
1
H
)
=
1
K and A
i
∈
B
(
ℋ
) are self-adjoint operators with spectra Sp
(
A
i
)
⊆
ℐ. The operator inequality of (2) was formulated for convex (without operator) continuous functions in [4] assuming the spectral conditions: Sp
(
A
)
⊆
[
a
,
b
] and Sp
(
A
i
)
∩
(
a
,
b
)
=
∅ for all A
i, where A
=
∑
i
=
1
n
‍
Φ
i
(
A
i
).Including positive operatorsP
i
∈
B
(
ℋ
) satisfying ∑
i
=
1
n
‍
Φ
i
(
P
i
)
=
1
K, we have that every convex continuous function f
:
ℐ
→
ℝ satisfies the inequality
(3)
f
(
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
)
≤
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
f
(
A
i
)
P
i
1
/
2
)
if provided the spectral conditions: Sp
(
A
)
⊆
[
a
,
b
] and Sp
(
A
i
)
∩
(
a
,
b
)
=
∅ for all self-adjoint operators A
i, and the operator sum A
=
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
). The inequality in (3) is possible because the operators P
i
1
/
2
A
i
P
i
1
/
2 and P
i
1
/
2
f
(
A
i
)
P
i
1
/
2 are self-adjoint.


## 2. Discrete and Operator Inequalities for Convex Functions and Trinomial Affine Combinations

### 2.1. Discrete Variants

Every numberx
∈
ℝ can be uniquely presented as the binomial affine combination
(4)
x
=
b
-
x
b
-
a
a
+
x
-
a
b
-
a
b
which is convex if and only if the number x belongs to the interval [
a
,
b
]. Given the function f
:
ℝ
→
ℝ, let f
{
a
,
b
}
line
:
ℝ
→
ℝ be the function of the chord line passing through the points A
(
a
,
f
(
a
)
) and B
(
b
,
f
(
b
)
) of the graph of f. Applying the affinity of f
{
a
,
b
}
line to the combination in (4), we get
(5)
f
{
a
,
b
}
line
(
x
)
=
b
-
x
b
-
a
f
(
a
)
+
x
-
a
b
-
a
f
(
b
)
.
If the function f is convex, then we have the inequality
(6)
f
(
x
)
≤
f
{
a
,
b
}
line
(
x
)
if
x
∈
[
a
,
b
]
,
and the reverse inequality
(7)
f
(
x
)
≥
f
{
a
,
b
}
line
(
x
)
if
x
∉
(
a
,
b
)
.Letα
,
β
,
γ
∈
ℝ be coefficients such that α
+
β
-
γ
=
1. Let a
,
b
,
c
∈
ℝ be points where a
<
b. We consider the affine combination α
a
+
β
b
-
γ
c. Inserting the affine combination c
=
λ
a
+
μ
b assuming that λ
+
μ
=
1, we get the binomial form
(8)
α
a
+
β
b
-
γ
c
=
(
α
-
γ
λ
)
a
+
(
β
-
γ
μ
)
b
.Lemma 1.
Letα
,
β
,
γ
∈
[
0,1
] be coefficients such that α
+
β
-
γ
=
1. Let a
,
b
,
c
∈
ℝ be points such that a
<
b and c
∈
[
a
,
b
].
Then the affine combination(9)
α
a
+
β
b
-
γ
c
∈
[
a
,
b
]
,
and every convex function f
:
[
a
,
b
]
→
ℝ satisfies the inequality
(10)
f
(
α
a
+
β
b
-
γ
c
)
≤
α
f
(
a
)
+
β
f
(
b
)
-
γ
f
(
c
)
.Proof.
The conditionc
=
λ
a
+
μ
b
∈
[
a
,
b
] involves λ
,
μ
∈
[
0,1
]. Then the binomial combination of the right-hand side in (8) is convex since its coefficients α
-
γ
λ
≥
α
-
γ
=
1
-
β
≥
0 and also β
-
γ
μ
≥
0. So, the combination α
a
+
β
b
-
γ belongs to [
a
,
b
]. Applying the inequality in (6) and the affinity of f
{
a
,
b
}
line, we get
(11)
f
(
α
a
+
β
b
-
γ
c
)
≤
f
{
a
,
b
}
line
(
α
a
+
β
b
-
γ
c
)
=
α
f
(
a
)
+
β
f
(
b
)
-
γ
f
{
a
,
b
}
line
(
c
)
≤
α
f
(
a
)
+
β
f
(
b
)
-
γ
f
(
c
)
because f
{
a
,
b
}
line
(
c
)
≥
f
(
c
).Lemma1 is trivially true if a
=
b. It is also valid for γ
∈
[
-
1,1
] because then the observed affine combinations with γ
≤
0 become convex, and associated inequalities follow from Jensen’s inequality. The similar combinations including γ
∈
[
-
1,1
] were observed in [5, Corollary 11 and Theorem 12] additionally using a monotone function g. If α
=
β
=
γ
=
1, then the inequality in (10) is reduced to simple Mercer’s variant of Jensen’s inequality obtained in [6].Lemma 2.
Letα
,
β
,
γ
∈
[
1
,
∞
) be coefficients such that α
+
β
-
γ
=
1. Let a
,
b
,
c
∈
ℝ be points such that a
<
b and c
∉
(
a
,
b
).
Then the affine combination(12)
α
a
+
β
b
-
γ
c
∉
(
a
,
b
)
,
and every convex function f
:
ℐ
→
ℝ, where ℐ
=
conv
⁡
{
a
,
b
,
c
} satisfies the inequality
(13)
f
(
α
a
+
β
b
-
γ
c
)
≥
α
f
(
a
)
+
β
f
(
b
)
-
γ
f
(
c
)
.Proof.
The conditionc
=
λ
a
+
μ
b
∉
(
a
,
b
) entails λ
≤
0 or λ
≥
1, and the coefficients of the binomial form of (8) satisfy α
-
γ
λ
≥
α
≥
1 if λ
≤
0, or α
-
γ
λ
≤
α
-
γ
=
1
-
β
≤
0 if λ
≥
1. So, the combination α
a
+
β
b
-
γ does not belong to (
a
,
b
). Applying the inequality in (7), we get the series of inequalities as in (11) but with the reverse inequality signs.It is not necessary to requireγ
∈
[
1
,
∞
) in Lemma 2, because it follows from the other coefficient conditions.


### 2.2. Operator Variants

We writeA
≤
B for self-adjoint operators A
,
B
∈
ℬ
(
ℋ
) if the inner product inequality 〈
A
x
,
x
〉
≤
〈
B
x
,
x
〉 holds for every vector x
∈
ℋ. A self-adjoint operator A is positive (nonnegative) if it is greater than or equal to null operator (A
≥
0). If Sp
(
A
)
⊆
ℐ and f
,
g
:
ℐ
→
ℝ are continuous functions such that f
(
x
)
≤
g
(
x
) for every x
∈
Sp
(
A
), then the operator inequality f
(
A
)
≤
g
(
A
) is valid. The bounds of a self-adjoint operator A are defined with
(14)
a
A
=
inf
⁡
∥
x
∥
=
1
〈
A
x
,
x
〉
,
b
A
=
sup
⁡
∥
x
∥
=
1
〈
A
x
,
x
〉
,
and its spectrum Sp
(
A
) is contained in [
a
A
,
b
A
] wherein we have the operator inequality
(15)
a
A
1
H
≤
A
≤
b
A
1
H
.More details on the theory of bounded operators and their inequalities can be found in [7]. The operator versions of Lemmas 1 and 2 follow.Corollary 3.
Letα
,
β
,
γ
∈
[
0,1
] be coefficients such that α
+
β
-
γ
=
1. Let A
∈
B
(
ℋ
) be a self-adjoint operator such that S
p
(
A
)
⊆
[
a
,
b
].
Then(16)
S
p
(
α
a
1
H
+
β
b
1
H
-
γ
A
)
⊆
[
a
,
b
]
,
and every convex continuous function f
:
[
a
,
b
]
→
ℝ satisfies the inequality
(17)
f
(
α
a
1
H
+
β
b
1
H
-
γ
A
)
≤
α
f
(
a
)
1
H
+
β
f
(
b
)
1
H
-
γ
f
(
A
)
.Proof.
The spectral inclusion in (16) follows from the inclusion in (9). Using the affinity of the function f
{
a
,
b
}
line and the operator inequalities f
{
a
,
b
}
line
(
·
)
≥
f
(
·
), we can replace the discrete inequalities in (11) with the operator inequalities.Corollary 4.
Letα
,
β
,
γ
∈
[
1
,
∞
) be coefficients such that α
+
β
-
γ
=
1. Let A
∈
B
(
ℋ
) be a self-adjoint operator such that S
p
(
A
)
∩
(
a
,
b
)
=
∅.
Then(18)
S
p
(
α
a
1
H
+
β
b
1
H
-
γ
A
)
∩
(
a
,
b
)
=
∅
,
and every convex continuous function f
:
ℐ
→
ℝ, where ℐ contains S
p
(
A
) and [
a
,
b
], satisfies the inequality
(19)
f
(
α
a
1
H
+
β
b
1
H
-
γ
A
)
≥
α
f
(
a
)
1
H
+
β
f
(
b
)
1
H
-
γ
f
(
A
)
.


## 2.1. Discrete Variants

Every numberx
∈
ℝ can be uniquely presented as the binomial affine combination
(4)
x
=
b
-
x
b
-
a
a
+
x
-
a
b
-
a
b
which is convex if and only if the number x belongs to the interval [
a
,
b
]. Given the function f
:
ℝ
→
ℝ, let f
{
a
,
b
}
line
:
ℝ
→
ℝ be the function of the chord line passing through the points A
(
a
,
f
(
a
)
) and B
(
b
,
f
(
b
)
) of the graph of f. Applying the affinity of f
{
a
,
b
}
line to the combination in (4), we get
(5)
f
{
a
,
b
}
line
(
x
)
=
b
-
x
b
-
a
f
(
a
)
+
x
-
a
b
-
a
f
(
b
)
.
If the function f is convex, then we have the inequality
(6)
f
(
x
)
≤
f
{
a
,
b
}
line
(
x
)
if
x
∈
[
a
,
b
]
,
and the reverse inequality
(7)
f
(
x
)
≥
f
{
a
,
b
}
line
(
x
)
if
x
∉
(
a
,
b
)
.Letα
,
β
,
γ
∈
ℝ be coefficients such that α
+
β
-
γ
=
1. Let a
,
b
,
c
∈
ℝ be points where a
<
b. We consider the affine combination α
a
+
β
b
-
γ
c. Inserting the affine combination c
=
λ
a
+
μ
b assuming that λ
+
μ
=
1, we get the binomial form
(8)
α
a
+
β
b
-
γ
c
=
(
α
-
γ
λ
)
a
+
(
β
-
γ
μ
)
b
.Lemma 1.
Letα
,
β
,
γ
∈
[
0,1
] be coefficients such that α
+
β
-
γ
=
1. Let a
,
b
,
c
∈
ℝ be points such that a
<
b and c
∈
[
a
,
b
].
Then the affine combination(9)
α
a
+
β
b
-
γ
c
∈
[
a
,
b
]
,
and every convex function f
:
[
a
,
b
]
→
ℝ satisfies the inequality
(10)
f
(
α
a
+
β
b
-
γ
c
)
≤
α
f
(
a
)
+
β
f
(
b
)
-
γ
f
(
c
)
.Proof.
The conditionc
=
λ
a
+
μ
b
∈
[
a
,
b
] involves λ
,
μ
∈
[
0,1
]. Then the binomial combination of the right-hand side in (8) is convex since its coefficients α
-
γ
λ
≥
α
-
γ
=
1
-
β
≥
0 and also β
-
γ
μ
≥
0. So, the combination α
a
+
β
b
-
γ belongs to [
a
,
b
]. Applying the inequality in (6) and the affinity of f
{
a
,
b
}
line, we get
(11)
f
(
α
a
+
β
b
-
γ
c
)
≤
f
{
a
,
b
}
line
(
α
a
+
β
b
-
γ
c
)
=
α
f
(
a
)
+
β
f
(
b
)
-
γ
f
{
a
,
b
}
line
(
c
)
≤
α
f
(
a
)
+
β
f
(
b
)
-
γ
f
(
c
)
because f
{
a
,
b
}
line
(
c
)
≥
f
(
c
).Lemma1 is trivially true if a
=
b. It is also valid for γ
∈
[
-
1,1
] because then the observed affine combinations with γ
≤
0 become convex, and associated inequalities follow from Jensen’s inequality. The similar combinations including γ
∈
[
-
1,1
] were observed in [5, Corollary 11 and Theorem 12] additionally using a monotone function g. If α
=
β
=
γ
=
1, then the inequality in (10) is reduced to simple Mercer’s variant of Jensen’s inequality obtained in [6].Lemma 2.
Letα
,
β
,
γ
∈
[
1
,
∞
) be coefficients such that α
+
β
-
γ
=
1. Let a
,
b
,
c
∈
ℝ be points such that a
<
b and c
∉
(
a
,
b
).
Then the affine combination(12)
α
a
+
β
b
-
γ
c
∉
(
a
,
b
)
,
and every convex function f
:
ℐ
→
ℝ, where ℐ
=
conv
⁡
{
a
,
b
,
c
} satisfies the inequality
(13)
f
(
α
a
+
β
b
-
γ
c
)
≥
α
f
(
a
)
+
β
f
(
b
)
-
γ
f
(
c
)
.Proof.
The conditionc
=
λ
a
+
μ
b
∉
(
a
,
b
) entails λ
≤
0 or λ
≥
1, and the coefficients of the binomial form of (8) satisfy α
-
γ
λ
≥
α
≥
1 if λ
≤
0, or α
-
γ
λ
≤
α
-
γ
=
1
-
β
≤
0 if λ
≥
1. So, the combination α
a
+
β
b
-
γ does not belong to (
a
,
b
). Applying the inequality in (7), we get the series of inequalities as in (11) but with the reverse inequality signs.It is not necessary to requireγ
∈
[
1
,
∞
) in Lemma 2, because it follows from the other coefficient conditions.


## 2.2. Operator Variants

We writeA
≤
B for self-adjoint operators A
,
B
∈
ℬ
(
ℋ
) if the inner product inequality 〈
A
x
,
x
〉
≤
〈
B
x
,
x
〉 holds for every vector x
∈
ℋ. A self-adjoint operator A is positive (nonnegative) if it is greater than or equal to null operator (A
≥
0). If Sp
(
A
)
⊆
ℐ and f
,
g
:
ℐ
→
ℝ are continuous functions such that f
(
x
)
≤
g
(
x
) for every x
∈
Sp
(
A
), then the operator inequality f
(
A
)
≤
g
(
A
) is valid. The bounds of a self-adjoint operator A are defined with
(14)
a
A
=
inf
⁡
∥
x
∥
=
1
〈
A
x
,
x
〉
,
b
A
=
sup
⁡
∥
x
∥
=
1
〈
A
x
,
x
〉
,
and its spectrum Sp
(
A
) is contained in [
a
A
,
b
A
] wherein we have the operator inequality
(15)
a
A
1
H
≤
A
≤
b
A
1
H
.More details on the theory of bounded operators and their inequalities can be found in [7]. The operator versions of Lemmas 1 and 2 follow.Corollary 3.
Letα
,
β
,
γ
∈
[
0,1
] be coefficients such that α
+
β
-
γ
=
1. Let A
∈
B
(
ℋ
) be a self-adjoint operator such that S
p
(
A
)
⊆
[
a
,
b
].
Then(16)
S
p
(
α
a
1
H
+
β
b
1
H
-
γ
A
)
⊆
[
a
,
b
]
,
and every convex continuous function f
:
[
a
,
b
]
→
ℝ satisfies the inequality
(17)
f
(
α
a
1
H
+
β
b
1
H
-
γ
A
)
≤
α
f
(
a
)
1
H
+
β
f
(
b
)
1
H
-
γ
f
(
A
)
.Proof.
The spectral inclusion in (16) follows from the inclusion in (9). Using the affinity of the function f
{
a
,
b
}
line and the operator inequalities f
{
a
,
b
}
line
(
·
)
≥
f
(
·
), we can replace the discrete inequalities in (11) with the operator inequalities.Corollary 4.
Letα
,
β
,
γ
∈
[
1
,
∞
) be coefficients such that α
+
β
-
γ
=
1. Let A
∈
B
(
ℋ
) be a self-adjoint operator such that S
p
(
A
)
∩
(
a
,
b
)
=
∅.
Then(18)
S
p
(
α
a
1
H
+
β
b
1
H
-
γ
A
)
∩
(
a
,
b
)
=
∅
,
and every convex continuous function f
:
ℐ
→
ℝ, where ℐ contains S
p
(
A
) and [
a
,
b
], satisfies the inequality
(19)
f
(
α
a
1
H
+
β
b
1
H
-
γ
A
)
≥
α
f
(
a
)
1
H
+
β
f
(
b
)
1
H
-
γ
f
(
A
)
.


## 3. Main Results

We want to extend and generalize the inequalities in (17) and (19) including positive operators and positive linear mappings. The main results are Theorems 8 and 9.Lemma 5.
LetΦ
i
:
B
(
ℋ
)
→
B
(
𝒦
) be linear mappings and let P
i
∈
B
(
ℋ
) be positive linear operators so that ∑
i
=
1
n
‍
Φ
i
(
P
i
)
=
1
K. Let A
i
∈
B
(
ℋ
) be self-adjoint operators.
Then every affine functiong
(
x
)
=
u
x
+
v, where u and v are real constants, satisfies the operator equality
(20)
g
(
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
)
=
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
g
(
A
i
)
P
i
1
/
2
)
.Proof.
Applying the affinity of the functiong and the assumption ∑
i
=
1
n
‍
Φ
i
(
P
i
)
=
1
K, it follows that
(21)
g
(
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
)
=
u
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
+
v
1
K
=
u
∑
i
=
1
n
‍
Φ
i
(
P
i
A
i
)
+
v
∑
i
=
1
n
Φ
i
(
P
i
)
‍
=
∑
i
=
1
n
Φ
i
(
P
i
1
/
2
(
u
A
i
+
v
1
H
)
P
i
1
/
2
)
‍
=
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
g
(
A
i
)
P
i
1
/
2
)
achieving the equality in (20).Lemma 6.
LetΦ
i
:
B
(
ℋ
)
→
B
(
𝒦
) be positive linear mappings and let P
i
∈
B
(
ℋ
) be positive linear operators so that ∑
i
=
1
n
‍
Φ
i
(
P
i
)
=
1
K. Let A
i
∈
B
(
ℋ
) be self-adjoint operators such that S
p
(
A
i
)
⊆
[
a
,
b
].
Then the spectrum of the operator sumA
=
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
) is contained in [
a
,
b
].Proof.
Applying the positive operatorsP
i and the positive mappings Φ
i to the assumed spectral inequalities
(22)
a
1
H
≤
A
i
≤
b
1
H
,
we get
(23)
a
Φ
i
(
P
i
)
≤
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
≤
b
Φ
i
(
P
i
)
.
Summing the above inequalities and using the assumption ∑
i
=
1
n
‍
Φ
i
(
P
i
)
=
1
K, we have
(24)
a
1
K
≤
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
≤
b
1
K
which provides that Sp
(
A
)
⊆
[
a
,
b
].Corollary 7.
LetΦ
i
:
B
(
ℋ
)
→
B
(
𝒦
) be positive linear mappings and let P
i
∈
B
(
ℋ
) be positive linear operators so that ∑
i
=
1
n
‍
Φ
i
(
P
i
)
=
1
K. Let A
i
∈
B
(
ℋ
) be self-adjoint operators such that S
p
(
A
i
)
⊆
[
a
,
b
].
Then every convex continuous functionf
:
[
a
,
b
]
→
ℝ satisfies the inequality
(25)
max
⁡
{
f
(
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
)
,
∑
i
=
1
n
Φ
i
(
P
i
1
/
2
f
(
A
i
)
P
i
1
/
2
)
‍
}
≤
f
{
a
,
b
}
l
i
n
e
(
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
)
≤
max
⁡
{
f
(
a
)
1
K
,
f
(
b
)
1
K
}
.Proof.
The inequality in (25) is the consequence of Lemmas 5 and 6, and the discrete inequality
(26)
max
⁡
{
f
(
∑
i
=
1
n
‍
p
i
x
i
)
,
∑
i
=
1
n
‍
p
i
f
(
x
i
)
}
≤
f
{
a
,
b
}
line
(
∑
i
=
1
n
‍
p
i
x
i
)
≤
max
⁡
{
f
(
a
)
,
f
(
b
)
}
,
where x
i
∈
[
a
,
b
] are points and p
i
∈
[
0,1
] are coefficients of the sum equal to 1.Theorem 8.
Letα
,
β
,
γ
∈
[
0,1
] be coefficients such that α
+
β
-
γ
=
1. Let Φ
i
:
B
(
ℋ
)
→
B
(
𝒦
) be positive linear mappings and let P
i
∈
B
(
ℋ
) be positive linear operators so that ∑
i
=
1
n
‍
Φ
i
(
P
i
)
=
1
K. Let A
i
∈
B
(
ℋ
) be self-adjoint operators such that S
p
(
A
i
)
⊆
[
a
,
b
].
Then the spectrum of the operator(27)
A
-
=
α
a
1
K
+
β
b
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
is contained in [
a
,
b
], and every convex continuous function f
:
[
a
,
b
]
→
ℝ satisfies the inequality
(28)
f
(
A
-
)
≤
α
f
(
a
)
1
K
+
β
f
(
b
)
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
f
(
A
i
)
P
i
1
/
2
)
.
If the functionf is concave, then the reverse inequality is valid in (28).Proof.
Taking the operator sumA
=
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
), the spectral inclusion Sp
(
A
-
)
⊆
[
a
,
b
] follows from Lemma 6 and the inclusion in (16). Assuming and applying the convexity of f and the affinity of f
{
a
,
b
}
line according to Lemma 5, we get
(29)
f
(
A
-
)
≤
f
{
a
,
b
}
line
(
α
a
1
K
+
β
b
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
)
=
α
f
(
a
)
1
K
+
β
f
(
b
)
1
K
-
γ
f
{
a
,
b
}
line
(
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
)
=
α
f
(
a
)
1
K
+
β
f
(
b
)
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
f
{
a
,
b
}
line
(
A
i
)
P
i
1
/
2
)
≤
α
f
(
a
)
1
K
+
β
f
(
b
)
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
f
(
A
i
)
P
i
1
/
2
)
because f
{
a
,
b
}
line
(
A
i
)
≥
f
(
A
i
).The version of Theorem8 for α
=
β
=
γ
=
1 and all P
i
=
1
H was obtained in [8] as the main result.Theorem 9.
Letα
,
β
,
γ
∈
[
1
,
∞
) be coefficients such that α
+
β
-
γ
=
1. Let Φ
i
:
B
(
ℋ
)
→
B
(
𝒦
) be positive linear mappings and let P
i
∈
B
(
ℋ
) be positive linear operators so that ∑
i
=
1
n
‍
Φ
i
(
P
i
)
=
1
K. Let A
i
∈
B
(
ℋ
) be self-adjoint operators such that S
p
(
A
i
)
∩
(
a
,
b
)
=
∅, and let A
=
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
) be the operator sum such that S
p
(
A
)
∩
(
a
,
b
)
=
∅.
Then the spectrum of the operator(30)
A
-
=
α
a
1
K
+
β
b
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
satisfies the relation S
p
(
A
-
)
∩
(
a
,
b
)
=
∅, and every convex continuous function f
:
ℐ
→
ℝ, where ℐ contains all spectra and [
a
,
b
], satisfies the inequality
(31)
f
(
A
-
)
≥
α
f
(
a
)
1
K
+
β
f
(
b
)
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
f
(
A
i
)
P
i
1
/
2
)
.
If the function f is concave, then the reverse inequality is valid in (31).Proof.
The relationSp
(
A
-
)
∩
(
a
,
b
)
=
∅ is the consequence of the relation in (18). Assuming and using the convexity of f and the affinity of f
{
a
,
b
}
line, as well as the inequalities f
{
a
,
b
}
line
(
A
i
)
≤
f
(
A
i
), we get the series of inequalities as in (29) but with the reverse inequality signs.


## 4. Application to Quasi-Arithmetic Means

In applications of convexity to quasi-arithmetic means, we use strictly monotone continuous functionsφ
,
ψ
:
ℐ
→
ℝ such that the function ψ
∘
φ
-
1 is convex, in which case we say that ψ is φ-convex. A similar notation is used for concavity. This terminology is taken from [9, Definition 1.19].A continuous functionf
:
ℐ
→
ℝ is said to be operator increasing on ℐ if A
≤
B implies f
(
A
)
≤
f
(
B
) for every pair of self-adjoint operators A
,
B
∈
B
(
ℋ
) with spectra in ℐ. A function f is said to be operator decreasing if the function -
f is operator increasing.Take an operator affine combination(32)
A
-
=
α
a
1
K
+
β
b
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
as in Theorem 8. If φ
:
[
a
,
b
]
→
ℝ is a strictly monotone continuous function, we define the φ-quasi-arithmetic mean of the combination A
- as the operator
(33)
M
φ
(
A
-
)
=
φ
-
1
(
α
φ
(
a
)
1
K
+
β
φ
(
b
)
1
K
∑
i
=
1
n
‍
g
g
g
g
g
g
g
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
φ
(
A
i
)
P
i
1
/
2
)
)
.
The spectrum of the operator M
φ
(
A
-
) is contained in [
a
,
b
] because the spectrum of the operator
(34)
A
-
φ
=
α
φ
(
a
)
1
K
+
β
φ
(
b
)
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
φ
(
A
i
)
P
i
1
/
2
)
is contained in φ
(
[
a
,
b
]
). The quasi-arithmetic means defined in (33) are invariant with respect to the affinity; that is, the equality
(35)
M
u
φ
+
v
(
A
-
)
=
M
φ
(
A
-
)
holds for all pairs of real numbers u
≠
0 and v. Indeed, if ψ
(
x
)
=
u
φ
(
x
)
+
v, then
(36)
A
-
ψ
=
u
A
-
φ
+
v
1
K
,
ψ
-
1
(
x
)
=
φ
-
1
(
1
u
(
x
-
v
)
)
,
and therefore, it follows that
(37)
M
ψ
(
A
-
)
=
ψ
-
1
(
A
-
ψ
)
=
φ
-
1
(
1
u
(
A
-
ψ
-
v
1
K
)
)
=
φ
-
1
(
A
-
φ
)
=
M
φ
(
A
-
)
.The order of the pair of quasi-arithmetic meansM
φ and M
ψ depends on convexity of the function ψ
∘
φ
-
1 and monotonicity of the function ψ. Theorem 8 can be applied to operator means as follows.Corollary 10.
LetA
- be an affine combination as in (32) satisfying the assumptions of Theorem 8. Let φ
,
ψ
:
[
a
,
b
]
→
ℝ be strictly monotone continuous functions.
Ifψ is either φ-convex with operator increasing ψ
-
1 or φ-concave with operator decreasing ψ
-
1, then one has the inequality
(38)
M
φ
(
A
-
)
≤
M
ψ
(
A
-
)
.
Ifψ is either φ-convex with operator decreasing ψ
-
1 or φ-concave with operator increasing ψ
-
1, then one has the reverse inequality in (38).Proof.
Let us prove the case in whichψ is φ-convex with operator increasing ψ
-
1. Put [
c
,
d
]
=
φ
(
[
a
,
b
]
). Applying the inequality in (28) of Theorem 8 to the affine combination A
-
φ of (34) with Sp
(
A
-
φ
)
⊆
[
c
,
d
] and the convex function f
=
ψ
∘
φ
-
1
:
[
c
,
d
]
→
ℝ, we get
(39)
ψ
∘
φ
-
1
(
A
-
φ
)
≤
A
-
ψ
.
Assigning the increasing function ψ
-
1 to the above inequality, we attain
(40)
M
φ
(
A
-
)
=
φ
-
1
(
A
-
φ
)
≤
ψ
-
1
(
A
-
ψ
)
=
M
ψ
(
A
-
)
which finishes the proof.Using Corollary10 we get the following version of the harmonic-geometric-arithmetic mean inequality for operators.Corollary 11.
IfA
- is an affine operator combination as in (32) satisfying the assumptions of Theorem 8 with the addition that [
a
,
b
]
⊂
(
0
,
∞
), then one has the harmonic-geometric-arithmetic operator inequality
(41)
(
(
α
a
+
β
b
)
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
-
1
P
i
1
/
2
)
)
-
1
≤
ln
⁡
a
α
b
β
1
K
-
γ
exp
⁡
(
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
(
ln
⁡
A
i
)
P
i
1
/
2
)
)
≤
(
α
a
+
β
b
)
1
K
-
γ
∑
i
=
1
n
‍
Φ
i
(
P
i
1
/
2
A
i
P
i
1
/
2
)
.Proof.
To prove the left-hand side of the inequality in (41) we use the functions φ
(
x
)
=
ln
⁡
x and ψ
(
x
)
=
x
-
1. Then ψ
∘
φ
-
1
(
x
)
=
exp
⁡
(
-
x
) and ψ
-
1
(
x
)
=
x
-
1, so ψ is φ-convex and ψ
-
1
(
x
)
=
x
-
1 is operator decreasing. Applying Corollary 10 to this case, we have
(42)
M
ln
⁡
x
(
A
-
)
≥
M
x
-
1
(
A
-
)
.
To prove the right-hand side we use the functionsφ
(
x
)
=
ln
⁡
x and ψ
(
x
)
=
x. Then ψ
∘
φ
-
1
(
x
)
=
exp
⁡
x and ψ
-
1
(
x
)
=
x, so ψ is φ-convex and ψ
-
1
(
x
)
=
x
-
1 is operator increasing. Applying the inequality in (38), we get
(43)
M
ln
⁡
x
(
A
-
)
≤
M
x
(
A
-
)
.
The double inequality in (41) follows by connecting the inequalities in (42) and (43).Quasi-arithmetic operator means without applying operator convexity were also investigated in [4, 10].



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*Source: 382364-2014-02-09.xml*