Chapter 2
Inequalities of the Jensen Type
2.1 Introduction
Jensen’s type inequalities in their various settings ranging from discrete to
continuous case play an important role in different branches of Modern
Mathematics. A simple search in the MathSciNet database of the American
Mathematical Society with the key words “jensen” and “inequality” in the title
reveals that there are more than 300 items intimately devoted to this famous result.
However, the number of papers where this inequality is applied is a lot larger and
far more difficult to find. It can be a good project in itself for someone to write a
monograph devoted to Jensen’s inequality in its different forms and its applications
across Mathematics.
In the introductory chapter we have recalled a number of Jensen’s type
inequalities for convex and operator convex functions of selfadjoint operators
in Hilbert spaces. In this chapter we present some recent results obtained by the
author that deal with different aspects of this well-researched inequality than those
recently reported in the book [19]. They include but are not restricted to the operator
version of the Dragomir–Ionescu inequality, Slater’s type inequalities for operators
and its inverses, Jensen’s inequality for twice differentiable functions whose
second derivatives satisfy some upper and lower bounds conditions, Jensen’s type
inequalities for log-convex functions and for differentiable log-convex functions
and their applications to Ky Fan’s inequality.
Finally, some Hermite–Hadamard’s type inequalities for convex functions and
Hermite–Hadamard’s type inequalities for operator convex functions are presented
as well.
All the above results are exemplified for some classes of elementary functions of
interest such as the power function and the logarithmic function.
S.S. Dragomir, Operator Inequalities of the Jensen, Čebyšev and Grüss Type,
SpringerBriefs in Mathematics, DOI 10.1007/978-1-4614-1521-3 2,
© Silvestru Sever Dragomir 2012
15
16
2 Inequalities of the Jensen Type
2.2 Reverses of the Jensen Inequality
2.2.1 An Operator Version of the Dragomir–Ionescu Inequality
The following result holds:
Theorem 2.1 (Dragomir, 2008, [8]). Let I be an interval and f W I ! R be
a convex and differentiable function on I̊ (the interior of I / whose derivative f 0
is continuous on I̊. If A is a selfadjoint operators on the Hilbert space H with
Sp.A/ Œm; M I̊; then
˛
˝
˛
˝
.0 / hf .A/x; xi f .hAx; xi/ f 0 .A/Ax; x hAx; xi f 0 .A/x; x
(2.1)
for any x 2 H with kxk D 1:
Proof. Since f is convex and differentiable, we have that
f .t/ f .s/ f 0 .t/ .t s/
for any t; s 2 Œm; M :
Now, if we chose in this inequality s D hAx; xi 2 Œm; M for any x 2 H with
kxk D 1 since Sp.A/ Œm; M ; then we have
f .t/ f .hAx; xi/ f 0 .t/ .t hAx; xi/
(2.2)
for any t 2 Œm; M any x 2 H with kxk D 1:
If we fix x 2 H with kxk D 1 in (2.2) and apply property (P) then we get
˝
˛
hŒf .A/ f .hAx; xi/ 1H x; xi f 0 .A/ .A hAx; xi 1H / x; x
for each x 2 H with kxk D 1; which is clearly equivalent to the desired inequality
(2.1).
t
u
Corollary 2.2 (Dragomir, 2008, [8]).
Assume that f is as in Theorem 2.1. If Aj
are selfadjoint operators with Sp Aj Œm; M I̊, j 2 f1; : : : ; ng and xj 2
2
P
H; j 2 f1; : : : ; ng with n xj D 1, then
j D1
0
1
n
n
X
X
˝ ˛
˝
˛
.0 /
f Aj xj ; xj f @
Aj xj ; xj A
j D1
j D1
n
n
n
X
˝ 0 ˛ X
˝
˛ X
˝ 0 ˛
f Aj Aj xj ; xj Aj xj ; xj f Aj xj ; xj :
j D1
j D1
j D1
(2.3)
2.2 Reverses of the Jensen Inequality
17
Corollary 2.3 (Dragomir, 2008, [8]). Assume that f is as in Theorem 2.1. If Aj
are selfadjoint operators
P with Sp Aj Œm; M I̊, j 2 f1; : : : ; ng and pj 0;
j 2 f1; : : : ; ng with nj D1 pj D 1; then
.0 /
* n
X
0*
+
+1
n
X
pj f Aj x; x f @
pj Aj x; x A
j D1
* n
X
j D1
+
+ * n
n
X
X
0
pj f Aj Aj x; x pj Aj x; x pj f Aj x; x ; (2.4)
+
*
0
j D1
j D1
j D1
for each x 2 H with kxk D 1:
Remark 2.4. Inequality (2.4), in the scalar case, namely
.0 /
n
X
0
1
n
X
pj f xj f @
pj xj A
j D1
n
X
j D1
n
n
X
X
pj f 0 xj xj pj xj pj f 0 xj ;
j D1
j D1
(2.5)
j D1
where xj 2 I̊, j 2 f1; : : : ; ng; has been obtained by the first time in 1994 by
Dragomir and Ionescu, see [16].
2.2.2 Further Reverses
In applications would be perhaps more useful to find upper bounds for the quantity
hf .A/x; xi f .hAx; xi/;
x2H
with
kxk D 1;
that are in terms of the spectrum margins m; M and of the function f .
The following result may be stated:
Theorem 2.5 (Dragomir, 2008, [8]). Let I be an interval and f W I ! R be a
convex and differentiable function on I̊ (the interior of I / whose derivative f 0 is
continuous on I̊: If A is a selfadjoint operator on the Hilbert space H with Sp.A/ Œm; M I̊; then
18
2 Inequalities of the Jensen Type
.0 / hf .A/x; xi f .hAx; xi/
8
i1=2
h
1
2
2
ˆ
0
0
ˆ
.M
m/
.A/xk
.A/x;
xi
hf
kf
ˆ
<
2
ˆ
i1=2
h
ˆ1
:̂ .f 0 .M / f 0 .m// kAxk2 hAx; xi2
2
1
.M m/ f 0 .M / f 0 .m/
4
(2.6)
for any x 2 H with kxk D 1:
We also have the inequality
.0 / hf .A/x; xi f .hAx; xi/
1
.M m/ f 0 .M / f 0 .m/
4
8
1
0
0
0
0
2
ˆ
< ŒhM x Ax; Ax mxi hf .M /x f .A/x; f .A/x f .m/xi ;
ˇ
ˇ
:̂ ˇˇhAx; xi M Cm ˇˇ ˇˇhf 0 .A/x; xi f 0 .M /Cf 0 .m/ ˇˇ
2
2
1
.M m/ f 0 .M / f 0 .m/
4
(2.7)
for any x 2 H with kxk D 1:
Moreover, if m > 0 and f 0 .m/ > 0; then we also have
.0 / hf .A/x; xi f .hAx; xi/
8
0 .M /f 0 .m//
1
ˆ
< .Mpm/.f
hAx; xi hf 0 .A/x; xi;
0 .M /f 0 .m/
M
mf
4
:̂ pM pm pf 0 .M / pf 0 .m/ ŒhAx; xi hf 0 .A/x; xi 12
(2.8)
for any x 2 H with kxk D 1:
Proof. We use the following Grüss’ type result we obtained in [6]:
Let A be a selfadjoint operator on the Hilbert space .H I h:; :i/ and assume that
Sp.A/ Œm; M for some scalars m < M: If h and g are continuous on Œm; M and
WD mint 2Œm;M h.t/ and WD maxt 2Œm;M h.t/; then
jhh.A/g.A/x; xi hh.A/x; xi hg.A/x; xij
i1=2
h
1
. / kg.A/xk2 hg.A/x; xi2
2
1
. / . ı/
4
(2.9)
for each x 2 H with kxk D1; where ı WD mint 2Œm;M g.t/ and WD maxt 2Œm;M g.t/:
2.2 Reverses of the Jensen Inequality
19
Therefore, we can state that
˝
˛
˝
˛
Af 0 .A/x; x hAx; xi f 0 .A/x; x
h
2 ˝
˛2 i1=2
1
.M m/ f 0 .A/x f 0 .A/x; x
2
1
.M m/ f 0 .M / f 0 .m/
4
(2.10)
and
˝
˛
˝
˛
Af 0 .A/x; x hAx; xi f 0 .A/x; x
i1=2
h
1 0
f .M / f 0 .m/ kAxk2 hAx; xi2
2
1
.M m/ f 0 .M / f 0 .m/
4
(2.11)
for each x 2 H with kxk D 1; which together with (2.1) provide the desired
result (2.6).
On making use of the inequality obtained in [7]:
jhh.A/g.A/x; xi hh.A/x; xi hg.A/x; xij
1
. / . ı/
4
8
1
ˆ
< Œh x h.A/x; f .A/x xi hx g.A/x; g.A/x ıxi 2;
ˇ
ˇ
:̂ ˇˇhh.A/x; xi C ˇˇ ˇˇhg.A/x; xi Cı ˇˇ
2
2
(2.12)
for each x 2 H with kxk D 1; we can state that
˛
˝
˛
˝ 0
Af .A/x; x hAx; xi f 0 .A/x; x
1
.M m/ f 0 .M / f 0 .m/
4
8
1
ˆ
< ŒhM x Ax; Ax mxi hf 0 .M /x f 0 .A/x; f 0 .A/x f 0 .m/xi 2 ;
ˇ
ˇ
ˇˇ
0
0
:̂ ˇhAx; xi M Cm ˇ ˇˇhf 0 .A/x; xi f .M /Cf .m/ ˇˇ
2
2
for each x 2 H with kxk D 1; which together with (2.1) provide the desired
result (2.7).
Further, in order to prove the third inequality, we make use of the following result
of Grüss type obtained in [7]:
20
2 Inequalities of the Jensen Type
If and ı are positive, then
jhh.A/g.A/x; xi hh.A/x; xi hg.A/x; xij
8
1
/.ı/
ˆ
< . p
hh.A/x; xi hg.A/x; xi;
ı
4p
p p
1
p
:̂
ı Œhh.A/x; xi hg.A/x; xi 2
(2.13)
for each x 2 H with kxk D 1:
Now, on making use of (2.13) we can state that
˝
˛
˝
˛
Af 0 .A/x; x hAx; xi f 0 .A/x; x
8
1 .M m/.f 0 .M /f 0 .m//
ˆ
ˆ
< pM mf 0 .M /f 0 .m/ hAx; xi hf 0 .A/x; xi;
4
p
ˆ p
1
p p 0
:̂
M m
f .M / f 0 .m/ ŒhAx; xi hf 0 .A/x; xi 2
for each x 2 H with kxk D 1; which together with (2.1) provide the desired
result (2.8).
t
u
Corollary 2.6 (Dragomir, 2008,[8]). Assume that f is as in Theorem 2.5. If Aj
are selfadjoint operators with Sp Aj Œm; M I̊, j 2 f1; : : : ; ng, then
0
1
n
n
X
X
˝ ˛
˝
˛
f Aj xj ; xj f @
Aj xj ; xj A
.0 /
j D1
j D1
8
˝ 0 ˛2 1=2
Pn 1
ˆ
f 0 Aj xj 2 Pn
ˆ
.M
m/
f
A
x
;
x
;
ˆ
j
j
j
j D1
j D1
<2
ˆ1
˝
˛2 1=2
ˆ
Pn Aj xj 2 Pn
:̂ .f 0 .M / f 0 .m//
A
x
;
x
;
j j
j
j D1
j D1
2
1
.M m/ f 0 .M / f 0 .m/
(2.14)
4
for any xj 2 H; j 2 f1; : : : ; ng with
We also have the inequality
Pn
j D1
2
xj D 1:
0
1
n
n
X
X
˝ ˛
˝
˛
f Aj xj ; xj f @
Aj xj ; xj A
.0 /
j D1
j D1
1
.M m/ f 0 .M / f 0 .m/
4
2.2 Reverses of the Jensen Inequality
21
8"
#1
ˆ
n ˝
ˆ
˛ 2
P
ˆ
ˆ
M xj Aj x; Aj xj mxj
ˆ
ˆ
ˆ
j D1
ˆ
ˆ
ˆ
#1=2
"
ˆ
<
n ˝
˛
P
f 0 .M /xj f 0 Aj xj ; f 0 Aj xj f 0 .m/xj
;
ˆ
j
D1
ˆ
ˆ
ˆ
ˆ
ˇ
ˇˇ
ˇ
ˆ
ˆ
ˆ ˇˇ P
n ˝
n ˝
˛ M Cm ˇˇ ˇˇ P
˛ f 0 .M /Cf 0 .m/ ˇˇ
ˆ
ˆˇ
0
Aj xj ; xj 2 ˇ ˇ
f Aj xj ; xj ˇ
:̂ ˇ
2
ˇˇ
ˇ
j D1
j D1
1
.M m/ f 0 .M / f 0 .m/
4
2
P
for any xj 2 H; j 2 f1; : : : ; ng with nj D1 xj D 1:
Moreover, if m > 0 and f 0 .m/ > 0; then we also have
0
1
n
n
X
X
˝ ˛
˝
˛
.0 /
f Aj xj ; xj f @
Aj xj ; xj A
j D1
(2.15)
j D1
81
˝
˛ Pn ˝ 0 ˛
.M m/.f 0 .M /f 0 .m// Pn
ˆ
Aj xj ; xj ;
ˆ
j D1 Aj xj ; xj
j D1 f
ˆ 4 pM mf 0 .M /f 0 .m/
ˆ
ˆ
< p
p
p p 0
M m
f .M / f 0 .m/
ˆ
ˆ
ˆ
i1
h
ˆ
:̂ Pn ˝A x ; x ˛ Pn ˝f 0 A x ; x ˛ 2
j D1
j
j
j
j
j D1
for any xj 2 H; j 2 f1; : : : ; ng with
Pn
j D1
j
(2.16)
j
2
xj D 1:
The following corollary also holds:
Corollary 2.7 (Dragomir, 2008, [8]). Assume that f is as in Theorem 2.1. If Aj
are selfadjoint operators
P with Sp Aj Œm; M I̊, j 2 f1; : : : ; ng and pj 0;
j 2 f1; : : : ; ng with nj D1 pj D 1; then
0*
* n
+
+1
n
X
X
.0 /
pj f Aj x; x f @
pj Aj x; x A
j D1
j D1
8
2
+2 31=2
*
ˆ
n
n
ˆ
0 2
P
P
1
ˆ
ˆ
ˆ
.M m/ 4
pj f Aj x pj f 0 Aj x; x 5 ;
ˆ
ˆ
<2
j D1
j D1
2
*
+2 31=2
ˆ
ˆ
ˆ
n
n
P
P
1
ˆ
2
ˆ
ˆ .f 0 .M / f 0 .m// 4
pj Aj x pj Aj x; x 5 ;
:̂ 2
j D1
j D1
1
.M m/ f 0 .M / f 0 .m/
4
for any x 2 H with kxk D 1:
(2.17)
22
2 Inequalities of the Jensen Type
We also have the inequality
.0 /
* n
X
0*
+
+1
n
X
pj f Aj x; x f @
pj Aj x; x A
j D1
j D1
1
.M m/ f 0 .M / f 0 .m/
4
8"
#1
ˆ
n
ˆ
˝
˛ 2
P
ˆ
ˆ
pj M x Aj x; Aj x mx
ˆ
ˆ
ˆ
j D1
ˆ
ˆ
"
#1=2
ˆ
ˆ
<
n
˝ 0
˛
P
0
0
0
pj f .M /x f Aj x; f Aj x f .m/x
;
ˆ
j D1
ˆ
ˆ
ˆ
ˆ
ˇ*
ˇ ˇ*
ˇ
+
+
ˆ
ˆ
ˇ n
ˇˇ n
ˇ
ˆ
ˆˇ P
f 0 .M /Cf 0 .m/ ˇ
M Cm ˇ ˇ P
0
ˆ
A
x;
x
p
A
x;
x
p
f
ˇ
ˇ
ˇ
ˇ
j
j
j
j
:̂ ˇ
2 ˇˇ
2
ˇ
j D1
j D1
1
.M m/ f 0 .M / f 0 .m/
4
(2.18)
for any x 2 H with kxk D 1:
Moreover, if m > 0 and f 0 .m/ > 0; then we also have
0*
+
* n
+1
n
X
X
.0 /
pj f Aj x; x f @
pj Aj x; x A
j D1
j D1
8
E
E DP
1 .M m/.f 0 .M /f 0 .m// DPn
n
ˆ
ˆ
pM mf 0 .M /f 0 .m/
pj Aj x; x
pj f 0 Aj x; x ;
ˆ
j
D1
j
D1
ˆ
4
ˆ
ˆ
< p
p
p p 0
M m
f .M / f 0 .m/
ˆ
ˆ
ˆ
ˆ
Ei 12
E DP
ˆ hDPn
n
:̂ p A x; x
p f 0 A x; x
j
j D1
j
j D1
j
j
(2.19)
for any x 2 H with kxk D 1:
Remark 2.8. Some of the inequalities in Corollary 2.7 can be used to produce
reverse norm inequalities for the sum of positive operators in the case when the
convex function f is non-negative and monotonic non-decreasing on Œ0; M :
For instance, if we use inequality (2.17), then we have
1
0
n
n
X
X
@
A
.0 / pj f Aj f pj Aj j D1
j D1
1
.M m/ f 0 .M / f 0 .m/ :
4
(2.20)
2.3 Some Slater Type Inequalities
23
Moreover, if we use inequality (2.19), then we obtain
1
0
X
X
n
n
@
A
.0 / pj f Aj f pj Aj j D1
j D1
8
P
n
n
P
1 .M m/.f 0 .M /f 0 .m// ˆ
0
ˆ
ˆ
pM mf 0 .M /f 0 .m/ pj Aj pj f Aj ;
ˆ
<4
j D1
j D1
# 1
"
2
P
ˆ
P
n
n
p
ˆ
p p 0
ˆ p
M m
f .M / f 0 .m/ pj Aj pj f 0 Aj :
:̂
j D1
j D1
(2.21)
2.3 Some Slater Type Inequalities
2.3.1 Slater Type Inequalities for Functions of Real Variables
Suppose that I is an interval of real numbers with interior I̊ and f W I ! R is
a convex function on I . Then f is continuous on I̊ and has finite left and right
derivatives at each point of I̊. Moreover, if x; y 2I̊ and x < y; then f0 .x/ fC0 .x/ f0 .y/ fC0 .y/ which shows that both f0 and fC0 are non-decreasing
function on I̊. It is also known that a convex function must be differentiable except
for at most countably many points.
For a convex function f W I ! R, the sub-differential
of f denoted by @f is the
set of all functions ' W I ! Œ1; 1 such that ' I̊ R and
f .x/ f .a/ C .x a/ '.a/
for any x; a 2 I:
It is also well known that if f is convex on I; then @f is non-empty, f0 , fC0 2 @f
and if ' 2 @f , then
f0 .x/ ' .x/ fC0 .x/
for any x 2 I̊.
In particular, ' is a non-decreasing function.
If f is differentiable and convex on I̊, then @f D ff 0 g :
The following result is well known in the literature as the Slater inequality:
Theorem 2.9 (Slater, 1981, [28]). If f W I ! R is a non-increasing
(nonPn
2
I;
p
0
with
P
WD
p
>
0
and
decreasing)
convex
function,
x
i
i
n
i
i
D1
Pn
i D1 pi ' .xi / ¤ 0; where ' 2 @f; then
Pn
n
1 X
pi xi ' .xi /
Pi D1
:
pi f .xi / f
n
Pn i D1
i D1 pi ' .xi /
(2.22)
24
2 Inequalities of the Jensen Type
As pointed out in [5, p. 208], the monotonicity assumption for the derivative '
can be replaced with the condition
Pn
pi xi ' .xi /
Pi D1
2 I;
n
i D1 pi ' .xi /
(2.23)
which is more general and can hold for suitable points in I and for not necessarily
monotonic functions.
2.3.2 Some Slater Type Inequalities for Operators
The following result holds:
Theorem 2.10 (Dragomir, 2008, [9]). Let I be an interval and f W I ! R be
a convex and differentiable function on I̊ (the interior of I / whose derivative f 0 is
continuous on I̊: If A is a selfadjoint operator on the Hilbert space H with Sp.A/ Œm; M I̊ and f 0 .A/ is a positive definite operator on H then
hAf 0 .A/x; xi
hf .A/x; xi
0f
hf 0 .A/x; xi
hAf 0 .A/x; xi hAx; xi hf 0 .A/x; xi
hAf 0 .A/x; xi
f0
hf 0 .A/x; xi
hf 0 .A/x; xi
(2.24)
for any x 2 H with kxk D 1:
Proof. Since f is convex and differentiable on I̊, then we have that
f 0 .s/ .t s/ f .t/ f .s/ f 0 .t/ .t s/
(2.25)
for any t; s 2 Œm; M :
Now, if we fix t 2 Œm; M and apply property (P) for the operator A; then for any
x 2 H with kxk D 1 we have
˛
˝ 0
f .A/ .t 1H A/ x; x hŒf .t/ 1H f .A/ x; xi
˝
˛
f 0 .t/ .t 1H A/ x; x
(2.26)
for any t 2 Œm; M and any x 2 H with kxk D 1:
Inequality (2.26) is equivalent with
˝
˛ ˝
˛
t f 0 .A/x; x f 0 .A/Ax; x f .t/ hf .A/x; xi
f 0 .t/t f 0 .t/ hAx; xi
for any t 2 Œm; M any x 2 H with kxk D 1:
(2.27)
2.3 Some Slater Type Inequalities
25
Now, since A is selfadjoint with mI A MI and f 0 .A/ is positive definite,
then mf 0 .A/ Af 0 .A/ Mf 0 .A/; i.e. m hf 0 .A/x; xi hAf 0 .A/x; xi M hf 0 .A/x; xi for any x 2 H with kxk D 1; which shows that
t0 WD
hAf 0 .A/x; xi
2 Œm; M hf 0 .A/x; xi
for any x 2 H
with
kxk D 1:
Finally, if we put t D t0 in (2.27), then we get the desired result (2.24).
Remark 2.11. It is important to observe that, the condition that f 0 .A/ is a positive
definite operator on H can be replaced with the more general assumption that
hAf 0 .A/x; xi
2 I̊
hf 0 .A/x; xi
for any x 2 H
with
kxk D 1;
(2.28)
which may be easily verified for particular convex functions f:
Remark 2.12. Now, if the functions are concave on I̊ and condition (2.28) holds,
then we have the inequality
hAf 0 .A/x; xi
0 hf .A/x; xi f
hf 0 .A/x; xi
hAx; xi hf 0 .A/x; xi hAf 0 .A/x; xi
hAf 0 .A/x; xi
f0
hf 0 .A/x; xi
hf 0 .A/x; xi
(2.29)
for any x 2 H with kxk D 1:
2.3.3 Further Reverses
The following results that provide perhaps more useful upper bounds for the nonnegative quantity:
f
hAf 0 .A/x; xi
hf 0 .A/x; xi
hf .A/x; xi
for x 2 H
with
kxk D 1
can be stated:
Theorem 2.13 (Dragomir, 2008, [9]). Let I be an interval and f W I ! R be
a convex and differentiable function on I̊ (the interior of I / whose derivative f 0 is
continuous on I̊: Assume that A is a selfadjoint operator on the Hilbert space H
with Sp.A/ Œm; M I̊ and f 0 .A/ is a positive definite operator on H: If we
define
0
0
1
0 hAf .A/x; xi
f
;
B f ; AI x WD
hf 0 .A/x; xi
hf 0 .A/x; xi
26
2 Inequalities of the Jensen Type
then
hAf 0 .A/x; xi
hf .A/x; xi
.0 /f
hf 0 .A/x; xi
8
h
i1=2
1
2
2
ˆ
0
0
ˆ
.M
m/
.A/xk
.A/x;
xi
kf
hf
<2
B f 0 ; AI x i1=2
h
ˆ1
:̂ .f 0 .M / f 0 .m// kAxk2 hAx; xi2
2
0
1
.M m/ f .M / f 0 .m/ B f 0 ; AI x
(2.30)
4
and
.0 /f
hAf 0 .A/x; xi
hf 0 .A/x; xi
2
hf .A/x; xi
61
B f 0 ; AI x 4 .M m/ f 0 .M / f 0 .m/
4
8
3
1
ˆ
< ŒhM x Ax; Ax mxi hf 0 .M /x f 0 .A/x; f 0 .A/x f 0 .m/xi 2
7
ˇ
5;
ˇ
ˇˇ
:̂ ˇhAx; xi M Cm ˇ ˇˇhf 0 .A/x; xi f 0 .M /Cf 0 .m/ ˇˇ
2
2
1
.M m/ f 0 .M / f 0 .m/ B f 0 ; AI x
4
(2.31)
for any x 2 H with kxk D 1; respectively.
Moreover, if A is a positive definite operator, then
hAf 0 .A/x; xi
hf .A/x; xi
hf 0 .A/x; xi
B f 0 ; AI x
8
0 .M /f 0 .m//
1
ˆ
< .Mpm/.f
hAx; xi hf 0 .A/x; xi ;
M mf 0 .M /f 0 .m/
4
:̂ pM pm pf 0 .M /pf 0 .m/ ŒhAx; xihf 0 .A/x; xi 12
.0 /f
(2.32)
for any x 2 H with kxk D 1:
Proof. We use the following Grüss’ type result we obtained in [6]:
Let A be a selfadjoint operator on the Hilbert space .H I h:; :i/ and assume that
Sp.A/ Œm; M for some scalars m < M: If h and g are continuous on Œm; M and
WD mint 2Œm;M h.t/ and WD maxt 2Œm;M h.t/; then
2.3 Some Slater Type Inequalities
27
jhh.A/g.A/x; xi hh.A/x; xi hg.A/x; xij
i1=2
h
1
. / kg.A/xk2 hg.A/x; xi2
2
1
. / . ı/
4
(2.33)
for each x 2 H with kxk D1; where ı WD mint 2Œm;M g.t/ and WD maxt 2Œm;M g.t/:
Therefore, we can state that
˛
˝
˛
˝ 0
Af .A/x; x hAx; xi f 0 .A/x; x
h
2 ˝
˛2 i1=2
1
.M m/ f 0 .A/x f 0 .A/x; x
2
1
.M m/ f 0 .M / f 0 .m/
(2.34)
4
and
˝
˛
˝
˛
Af 0 .A/x; x hAx; xi f 0 .A/x; x
i1=2
h
1 f 0 .M / f 0 .m/ kAxk2 hAx; xi2
2
1
.M m/ f 0 .M / f 0 .m/
4
(2.35)
for each x 2 H with kxk D 1; which together with (2.24) provide the desired
result (2.30).
On making use of the inequality obtained in [7]
jhh.A/g.A/x; xi hh.A/x; xi hg.A/x; xij
1
. / . ı/
4
8
1
< Œh x h.A/x; f .A/x xi hx g.A/x; g.A/x ıxi 2;
ˇ
ˇˇ
: ˇhh.A/x; xi C ˇˇ ˇˇhg.A/x; xi Cı ˇˇ
2
(2.36)
2
for each x 2 H with kxk D 1; we can state that
˝ 0
˛
˝
˛
Af .A/x; x hAx; xi f 0 .A/x; x
1
.M m/ f 0 .M / f 0 .m/
4
8
1
< ŒhM x Ax; Ax mxi hf 0 .M /x f 0 .A/x; f 0 .A/x f 0 .m/xi 2;
ˇ
ˇ
ˇ
: ˇhAx; xi M Cm ˇˇ ˇˇhf 0 .A/x; xi f 0 .M /Cf 0 .m/ ˇˇ
2
2
28
2 Inequalities of the Jensen Type
for each x 2 H with kxk D 1; which together with (2.24) provide the desired
result (2.31).
Further, in order to prove the third inequality, we make use of the following result
of Grüss type obtained in [7]:
If and ı are positive, then
jhh.A/g.A/x; xi hh.A/x; xi hg.A/x; xij
8
1
/.ı/
ˆ
< . p
hh.A/x; xi hg.A/x; xi;
ı
4
p
p p
1
p
:̂
ı Œhh.A/x; xi hg.A/x; xi 2
for each x 2 H with kxk D 1:
Now, on making use of (2.37) we can state that
˝ 0
˛
˝
˛
Af .A/x; x hAx; xi f 0 .A/x; x
8
1 .M m/.f 0 .M /f 0 .m//
ˆ
ˆ
< pM mf 0 .M /f 0 .m/ hAx; xi hf 0 .A/x; xi ;
4
p
ˆ p
1
p p 0
:̂
M m
f .M / f 0 .m/ ŒhAx; xi hf 0 .A/x; xi 2
(2.37)
(2.38)
for each x 2 H with kxk D 1; which together with (2.24) provide the desired
result (2.32).
t
u
Remark 2.14. We observe, from the first inequality in (2.32), that
.1 /
1 .M m/ .f 0 .M / f 0 .m//
hAf 0 .A/x; xi
C1
p
4
hAx; xi hf 0 .A/x; xi
M mf 0 .M /f 0 .m/
which implies that
f
0
hAf 0 .A/x; xi
hf 0 .A/x; xi
"
f
0
#
!
1 .M m/ .f 0 .M / f 0 .m//
C 1 hAx; xi
p
4
M mf 0 .M /f 0 .m/
for each x 2 H with kxk D 1; since f 0 is monotonic non-decreasing and A is
positive definite.
Now, the first inequality in (2.32) implies the following result:
hAf 0 .A/x; xi
hf .A/x; xi
.0 /f
hf 0 .A/x; xi
1 .M m/ .f 0 .M / f 0 .m//
p
4
M mf 0 .M /f 0 .m/
#
!
"
1 .M m/ .f 0 .M / f 0 .m//
0
f
C 1 hAx; xi hAx; xi
p
4
M mf 0 .M /f 0 .m/
for each x 2 H with kxk D 1:
(2.39)
2.4 Other Inequalities for Convex Functions
29
From the second inequality in (2.32) we also have
.0 /f
hAf 0 .A/x; xi
hf 0 .A/x; xi
hf .A/x; xi
p
p
p p 0
M m
f .M / f 0 .m/
"
f
0
#
!
12
1 .M m/ .f 0 .M / f 0 .m//
hAx; xi
p
C 1 hAx; xi
4
hf 0 .A/x; xi
M mf 0 .M /f 0 .m/
(2.40)
for each x 2 H with kxk D 1:
Remark 2.15. If the condition that f 0 .A/ is a positive definite operator on H from
Theorem 2.13 is replaced by condition (2.28), then inequalities (2.30) and (2.33)
will still hold. Similar inequalities for concave functions can be stated. However,
the details are not provided here.
2.4 Other Inequalities for Convex Functions
2.4.1 Some Inequalities for Two Operators
The following result holds:
Theorem 2.16 (Dragomir, 2008, [10]). Let I be an interval and f W I ! R be
a convex and differentiable function on I̊ (the interior of I / whose derivative f 0 is
continuous on I̊: If A and B are selfadjoint operators on the Hilbert space H with
Sp.A/; Sp.B/ Œm; M I̊; then
˝
˛
˛
˝
f 0 .A/x; x hBy; yi f 0 .A/Ax; x
hf .B/y; yi hf .A/x; xi
˝
˝
˛
˛
f 0 .B/By; y hAx; xi f 0 .B/y; y
(2.41)
for any x; y 2 H with kxk D kyk D 1:
In particular, we have
˝
˛
˝
˛
f 0 .A/x; x hAy; yi f 0 .A/Ax; x
hf .A/y; yi hf .A/x; xi
˝
˝
˛
˛
f 0 .A/Ay; y hAx; xi f 0 .A/y; y
for any x; y 2 H with kxk D kyk D 1 and
(2.42)
30
2 Inequalities of the Jensen Type
˝ 0
˛
˛
˝
f .A/x; x hBx; xi f 0 .A/Ax; x
hf .B/x; xi hf .A/x; xi
˝
˝
˛
˛
f 0 .B/Bx; x hAx; xi f 0 .B/x; x
(2.43)
for any x 2 H with kxk D 1:
Proof. Since f is convex and differentiable on I̊, then we have that
f 0 .s/ .t s/ f .t/ f .s/ f 0 .t/ .t s/
(2.44)
for any t; s 2 Œm; M :
Now, if we fix t 2 Œm; M and apply property (P) for the operator A; then for any
x 2 H with kxk D 1 we have
˛
˝ 0
f .A/ .t 1H A/ x; x hŒf .t/ 1H f .A/ x; xi
˝
˛
f 0 .t/ .t 1H A/ x; x
(2.45)
for any t 2 Œm; M and any x 2 H with kxk D 1:
Inequality (2.45) is equivalent with
˝
˛ ˝
˛
t f 0 .A/x; x f 0 .A/Ax; x f .t/ hf .A/x; xi
f 0 .t/t f 0 .t/ hAx; xi
(2.46)
for any t 2 Œm; M and any x 2 H with kxk D 1:
If we fix x 2 H with kxk D 1 in (2.46) and apply property (P) for the operator
B; then we get
˛
˝
˛ ˛
˝˝ 0
f .A/x; x B f 0 .A/Ax; x 1H y; y hŒf .B/ hf .A/x; xi 1H y; yi
˝
˛
f 0 .B/B hAx; xi f 0 .B/ y; y
for each y 2 H with kyk D 1; which is clearly equivalent to the desired inequality
(2.41).
t
u
Remark 2.17. If we fix x 2 H with kxk D 1 and choose B D hAx; xi 1H ; then we
obtain from the first inequality in (2.41) the reverse of the Mond–Pečarić inequality
obtained by the author in [8]. The second inequality will provide the Mond–Pečarić
inequality for convex functions whose derivatives are continuous.
The following corollary is of interest:
Corollary 2.18. Let I be an interval and f W I ! R be a convex and differentiable
function on I̊ whose derivative f 0 is continuous on I̊: Also, suppose that A is a
selfadjoint operator on the Hilbert space H with Sp.A/ Œm; M I̊: If g is
non-increasing and continuous on Œm; M and
f 0 .A/ Œg.A/ A 0
(2.47)
2.4 Other Inequalities for Convex Functions
31
in the operator order of B .H /; then
.f ı g/ .A/ f .A/
(2.48)
in the operator order of B .H /:
The following result may be stated as well:
Theorem 2.19 (Dragomir, 2008, [10]). Let I be an interval and f W I ! R be
a convex and differentiable function on I̊ (the interior of I / whose derivative f 0 is
continuous on I̊: If A and B are selfadjoint operators on the Hilbert space H with
Sp.A/; Sp.B/ Œm; M I̊; then
f 0 .hAx; xi/ .hBy; yi hAx; xi/
hf .B/y; yi f .hAx; xi/
˝
˛
˛
˝
f 0 .B/By; y hAx; xi f 0 .B/y; y
(2.49)
for any x; y 2 H with kxk D kyk D 1:
In particular, we have
f 0 .hAx; xi/ .hAy; yi hAx; xi/
hf .A/y; yi f .hAx; xi/
˝
˛
˛
˝
f 0 .A/Ay; y hAx; xi f 0 .A/y; y
(2.50)
for any x; y 2 H with kxk D kyk D 1 and
f 0 .hAx; xi/ .hBx; xi hAx; xi/ hf .B/x; xi f .hAx; xi/
˝
˛
˛
˝
f 0 .B/Bx; x hAx; xi f 0 .B/x; x
(2.51)
for any x 2 H with kxk D 1:
Proof. Since f is convex and differentiable on I̊, then we have that
f 0 .s/ .t s/ f .t/ f .s/ f 0 .t/ .t s/
(2.52)
for any t; s 2 Œm; M :
If we choose s D hAx; xi 2 Œm; M ; with a fix x 2 H with kxk D 1; then we
have
f 0 .hAx; xi/ .t hAx; xi/ f .t/ f .hAx; xi/
f 0 .t/ .t hAx; xi/
for any t 2 Œm; M :
(2.53)
32
2 Inequalities of the Jensen Type
Now, if we apply property (P) to inequality (2.53) and the operator B, then we get
˝
f 0 .hAx; xi/ .B hAx; xi 1H / y; yi
hŒf .B/ f .hAx; xi/ 1H y; yi
˝
˛
f 0 .B/ .B hAx; xi 1H / y; y
(2.54)
for any x; y 2 H with kxk D kyk D 1; which is equivalent with the desired
result (2.49).
Remark 2.20. We observe that if we choose B D A in (2.51) or y D x in (2.50)
then we recapture the Mond–Pečarić inequality and its reverse from (2.1).
The following particular case of interest follows from Theorem 2.19:
Corollary 2.21 (Dragomir, 2008, [10]). Assume that f; A and B are as in Theorem 2.19. If, either f is increasing on Œm; M and B A in the operator order of
B .H / or f is decreasing and B A; then we have the Jensen’s type inequality
hf .B/x; xi f .hAx; xi/
(2.55)
for any x 2 H with kxk D 1:
The proof is obvious by the first inequality in (2.51) and the details are omitted.
2.5 Some Jensen Type Inequalities for Twice Differentiable
Functions
2.5.1 Jensen’s Inequality for Twice Differentiable Functions
The following result may be stated:
Theorem 2.22 (Dragomir, 2008, [11]). Let A be a positive definite operator on
the Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M
with 0 < m < M: If f is a twice differentiable function on .m; M / and for p 2
.1; 0/ [ .1; 1/ we have for some < that
t 2p
f 00 .t/ p .p 1/
t 2 .m; M / ;
(2.56)
hAp x; xi hAx; xip hf .A/x; xi f .hAx; xi/
hAp x; xi hAx; xip
(2.57)
for any
then
for each x 2 H with kxk D 1:
2.5 Some Jensen Type Inequalities for Twice Differentiable Functions
33
If
ı
t 2p
f 00 .t/ p .1 p/
t 2 .m; M /
(2.58)
ı hAx; xip hAp x; xi hf .A/x; xi f .hAx; xi/
hAx; xip hAp x; xi
(2.59)
for any
and for some ı < ; where p 2 .0; 1/, then
for each x 2 H with kxk D 1:
Proof. Consider the function g ;p W .m; M / ! R given by g ;p .t/ D f .t/ t p
where p 2 .1; 0/ [ .1; 1/ : The function g ;p is twice differentiable,
g00;p .t/ D f 00 .t/ p .p 1/ t p2
for any t 2 .m; M / and by (2.56) we deduce that g ;p is convex on .m; M /: Now,
applying the Mond and Pečarić inequality for g ;p we have
0 h.f .A/ Ap / x; xi f .hAx; xi/ hAx; xip
D hf .A/x; xi f .hAx; xi/ hAp x; xi hAx; xip
which is equivalent with the first inequality in (2.57).
By defining the function g;p W .m; M / ! R given by g;p .t/ D t p f .t/
and applying the same argument we deduce the second part of (2.57).
The rest goes likewise and the details are omitted.
t
u
Remark 2.23. We observe that if f is a twice differentiable function on .m; M / and
' WD inft 2.m;M / f 00 .t/; Φ WD supt 2.m;M / f 00 .t/; then by (2.57) we get the inequality
i
˛
1 h˝ 2
' A x; x hAx; xi2 hf .A/x; xi f .hAx; xi/
2
i
˛
1 h˝
Φ A2 x; x hAx; xi2
2
(2.60)
for each x 2 H with kxk D 1:
We observe that inequality (2.60) holds for selfadjoint operators that are not
necessarily positive.
The next result provides some inequalities for the function f which replace the
cases p D 0 and p D 1 that were not allowed in Theorem 2.22.
34
2 Inequalities of the Jensen Type
Theorem 2.24 (Dragomir, 2008, [10]). Let A be a positive definite operator on the
Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with 0 <
m < M: If f is a twice differentiable function on .m; M / and we have for some
< that
t 2 f 00 .t/ for any
t 2 .m; M /;
(2.61)
then
.ln .hAx; xi/ hln Ax; xi/ hf .A/x; xi f .hAx; xi/
.ln .hAx; xi/ hln Ax; xi/
(2.62)
for each x 2 H with kxk D 1:
If
ı t f 00 .t/ for any
t 2 .m; M /
(2.63)
for some ı < , then
ı .hA ln Ax; xi hAx; xi ln .hAx; xi//
hf .A/x; xi f .hAx; xi/
.hA ln Ax; xi hAx; xi ln .hAx; xi//
(2.64)
for each x 2 H with kxk D 1:
Proof. Consider the function g ;0 W .m; M / ! R given by g ;0 .t/ D f .t/ C ln t:
The function g ;0 is twice differentiable,
g00;p .t/ D f 00 .t/ t 2
for any t 2 .m; M / and by (2.61) we deduce that g ;0 is convex on .m; M /: Now,
applying the Mond and Pečarić inequality for g ;0 we have
0 h.f .A/ C ln A/ x; xi Œf .hAx; xi/ C ln .hAx; xi/
D hf .A/x; xi f .hAx; xi/ Œln .hAx; xi/ hln Ax; xi
which is equivalent with the first inequality in (2.62).
By defining the function g;0 W .m; M / ! R given by g;0 .t/ D ln t f .t/
and applying the same argument we deduce the second part of (2.62).
The rest goes likewise for the functions
gı;1 .t/ D f .t/ ıt ln t
and the details are omitted.
and
g;0 .t/ D t ln t f .t/
2.6 Some Jensen’s Type Inequalities for Log-Convex Functions
35
2.6 Some Jensen’s Type Inequalities for Log-Convex Functions
2.6.1 Preliminary Results
The following result that provides an operator version for the Jensen inequality for
convex functions is due to Mond and Pečarić [25] (see also [19, p. 5]):
Theorem 2.25 (Mond–Pečarić, 1993, [25]). Let A be a selfadjoint operator on
the Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with
m < M: If f is a convex function on Œm; M ; then
f .hAx; xi/ hf .A/x; xi
(MP)
for each x 2 H with kxk D 1:
Taking into account the above result and its applications for various concrete
examples of convex functions, it is therefore natural to investigate the corresponding
results for the case of log-convex functions, namely functions f W I ! .0; 1/ for
which ln f is convex.
We observe that such functions satisfy the elementary inequality
f ..1 t/ a C tb/ Œf .a/1t Œf .b/t
(2.65)
for any a; b 2 I and t 2 Œ0; 1 : Also, due to the fact that the weighted geometric
mean is less than the weighted arithmetic mean, it follows that any log-convex
function is a convex function. However, obviously, there are functions that are
convex but not log-convex.
As an immediate consequence of the Mond–Pečarić inequality above, we can
provide the following result:
Theorem 2.26 (Dragomir, 2010, [14]). Let A be a selfadjoint operator on the
Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with
m < M: If g W Œm; M ! .0; 1/ is log-convex, then
g .hAx; xi/ exp hln g.A/x; xi hg.A/x; xi
(2.66)
for each x 2 H with kxk D 1:
Proof. Consider the function f WD ln g; which is convex on Œm; M : Writing (MP)
for f we get ln Œg .hAx; xi/ hln g.A/x; xi; for each x 2 H with kxk D 1; which,
by taking the exponential, produces the first inequality in (2.66).
If we also use (MP) for the exponential function, we get
exp hln g.A/x; xi hexp Œln g.A/ x; xi D hg.A/x; xi
for each x 2 H with kxk D 1 and the proof is complete.
36
2 Inequalities of the Jensen Type
The case of sequences of operators may be of interest and is embodied in the
following corollary:
Corollary 2.27 (Dragomir, 2010, [14]).
Assume that g is as in Theorem 2.26. If
Aj are selfadjoint operators with Sp Aj Œm; M , j 2 f1; : : : ; ng and xj 2
2
P
H; j 2 f1; : : : ; ng with nj D1 xj D 1, then
0
g@
n
X
˝
˛
1
Aj xj ; xj A exp
j D1
* n
X
ln g Aj xj ; xj
+
j D1
*
+
n
X
g Aj xj ; xj :
(2.67)
j D1
t
u
Proof. Follows from Theorem 2.26 and we omit the details.
In particular we have:
Corollary 2.28 (Dragomir, 2010, [14]).
Assume that g is as in Theorem 2.26. If Aj
are selfadjoint operators
with
Sp
A
Œm; M I̊, j 2 f1; : : : ; ng and pj 0;
j
P
j 2 f1; : : : ; ng with nj D1 pj D 1; then
0*
g@
n
X
+1
*
n
Y
pj
g Aj
pj Aj x; x A x; x
j D1
*
j D1
n
X
+
+
pj g Aj x; x
(2.68)
j D1
for each x 2 H with kxk D 1:
Proof. Follows from Corollary 2.27 by choosing xj D
where x 2 H with kxk D 1:
p
pj x; j 2 f1; : : : ; ng
The following reverse for the Mond–Pečarić inequality that generalizes the
scalar Lah–Ribarić inequality for convex functions is well known, see for instance [19, p.57]:
Theorem 2.29. Let A be a selfadjoint operator on the Hilbert space H and assume
that Sp.A/ Œm; M for some scalars m; M with m < M: If f is a convex function
on Œm; M ; then
hf .A/x; xi M hAx; xi
hAx; xi m
f .m/ C
f .M /
M m
M m
for each x 2 H with kxk D 1:
(2.69)
2.6 Some Jensen’s Type Inequalities for Log-Convex Functions
37
This result can be improved for log-convex functions as follows:
Theorem 2.30 (Dragomir, 2010, [14]). Let A be a selfadjoint operator on the
Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with
m < M: If g W Œm; M ! .0; 1/ is log-convex, then
E
Dh
M1H A
Am1H i
hg.A/x; xi Œg.m/ M m Œg.M / M m x; x
M hAx; xi
hAx; xi m
g.m/ C
g.M /
M m
M m
(2.70)
and
M hAx;xi
hAx;xim
g .hAx; xi/ Œg.m/ M m Œg.M / M m
E
Dh
M1H A
Am1H i
Œg.m/ M m Œg.M / M m x; x
(2.71)
for each x 2 H with kxk D 1:
Proof. Observe that, by the log-convexity of g; we have
M t
t m
g.t/ D g
mC
M
M m
M m
M t
t m
Œg.m/ M m Œg.M / M m
(2.72)
for any t 2 Œm; M :
Applying property (P) for the operator A, we have that
hg.A/x; xi hΨ.A/x; xi
M t
t m
for each x 2 H with kxk D 1; where Ψ.t/ WD Œg.m/ M m Œg.M / M m ; t 2 Œm; M :
This proves the first inequality in (2.70).
Now, observe that, by the weighted arithmetic mean–geometric mean inequality
we have
M t
t m
Œg.m/ M m Œg.M / M m t m
M t
g.m/ C
g.M /
M m
M m
for any t 2 Œm; M :
Applying property (P) for the operator A, we deduce the second inequality
in (2.70).
Further on, if we use inequality (2.72) for t D hAx; xi 2 Œm; M , then we deduce
the first part of (2.71).
Now, observe that the function Ψ introduced above can be rearranged to read as
g.M /
Ψ.t/ D g.m/
g.m/
Mt m
m
showing that Ψ is a convex function on Œm; M :
; t 2 Œm; M 38
2 Inequalities of the Jensen Type
Applying Mond–Pečarić’s inequality for Ψ we deduce the second part of (2.71)
and the proof is complete.
2.6.2 Jensen’s Inequality for Differentiable Log-Convex
Functions
The following result provides a reverse for the Jensen type inequality (MP):
Theorem 2.31 (Dragomir, 2008, [8]). Let J be an interval and f W J ! R be a
convex and differentiable function on J̊ (the interior of J / whose derivative f 0 is
continuous on J̊: If A is a selfadjoint operator on the Hilbert space H with Sp.A/ Œm; M J̊; then
.0 / hf .A/x; xi f .hAx; xi/
˝
˛
˝
˛
f 0 .A/Ax; x hAx; xi f 0 .A/x; x
(2.73)
for any x 2 H with kxk D 1:
The following result may be stated:
Proposition 2.32 (Dragomir, 2010, [14]). Let J be an interval and g W J ! R be
a differentiable log-convex function on J̊ whose derivative g 0 is continuous on J̊. If
A is a selfadjoint operator on the Hilbert space H with Sp.A/ Œm; M J̊; then
.1 /
hD
E
exp hln g.A/x; xi
exp g 0 .A/ Œg.A/1 Ax; x
g .hAx; xi/
D
Ei
hAx; xi g 0 .A/ Œg.A/1 x; x
(2.74)
for each x 2 H with kxk D 1:
Proof. It follows by inequality (2.73) written for the convex function f D ln g that
hln g.A/x; xi ln g .hAx; xi/
E
D
E
D
C g 0 .A/ Œg.A/1 Ax; x hAx; xi g0 .A/ Œg.A/1 x; x
for each x 2 H with kxk D 1:
Now, taking the exponential and dividing by g .hAx; xi/ > 0 for each x 2 H
with kxk D 1; we deduce the desired result (2.74).
u
t
The following result that provides both a refinement and a reverse of the
multiplicative version of Jensen’s inequality can be stated as well:
Theorem 2.33 (Dragomir, 2010, [14]). Let J be an interval and g W J ! R be a
log-convex differentiable function on J̊ whose derivative g 0 is continuous on J̊. If A
is a selfadjoint operator on the Hilbert space H with Sp.A/ Œm; M J̊; then
2.6 Some Jensen’s Type Inequalities for Log-Convex Functions
39
g 0 .hAx; xi/
.A hAx; xi 1H / x; x
1 exp
g .hAx; xi/
h
i
E
hg.A/x; xi D
exp g 0 .A/ Œg.A/1 .A hAx; xi 1H / x; x
g .hAx; xi/
(2.75)
for each x 2 H with kxk D 1; where 1H denotes the identity operator on H:
Proof. It is well known that if h W J ! R is a convex differentiable function on J̊,
then the following gradient inequality holds:
h.t/ h.s/ h0 .s/.t s/
for any t; s 2J̊.
Now, if we write this inequality for the convex function h D ln g; then we get
ln g.t/ ln g.s/ g 0 .s/
.t s/
g.s/
(2.76)
which is equivalent with
g 0 .s/
g.t/ g.s/ exp
.t s/
g.s/
(2.77)
for any t; s 2J̊.
Further, if we take s WD hAx; xi 2 Œm; M J̊; for a fixed x 2 H with kxk D 1;
in inequality (2.77), then we get
g.t/ g .hAx; xi/ exp
g 0 .hAx; xi/
.t hAx; xi/
g .hAx; xi/
for any t 2J̊.
Utilizing property (P) for the operator A and the Mond–Pečarić inequality for the
exponential function, we can state the following inequality that is of interest in itself
as well:
hg.A/y; yi
0
g .hAx; xi/
.A hAx; xi 1H / y; y
g .hAx; xi/ exp
g .hAx; xi/
0
g .hAx; xi/
.hAy; yi hAx; xi/
g .hAx; xi/ exp
g .hAx; xi/
(2.78)
for each x; y 2 H with kxk D kyk D 1:
Further, if we put y D x in (2.78), then we deduce the first and the second
inequality in (2.75).
40
2 Inequalities of the Jensen Type
Now, if we replace s with t in (2.77) we can also write the inequality
g.t/ exp
g 0 .t/
.s t/ g.s/
g.t/
which is equivalent with
g 0 .t/
g.t/ g.s/ exp
.t s/
g.t/
(2.79)
for any t; s 2J̊.
Further, if we take s WD hAx; xi 2 Œm; M J̊; for a fixed x 2 H with kxk D 1;
in inequality (2.79), then we get
g.t/ g .hAx; xi/ exp
g 0 .t/
.t hAx; xi/
g.t/
for any t 2J̊.
Utilizing property (P) for the operator A, then we can state the following
inequality that is of interest in itself as well:
hg.A/y; yi
D
h
i
E
g .hAx; xi/ exp g 0 .A/ Œg.A/1 .A hAx; xi 1H / y; y
(2.80)
for each x; y 2 H with kxk D kyk D 1:
Finally, if we put y D x in (2.80), then we deduce the last inequality in
(2.75).
u
t
The following reverse inequality may be proven as well:
Theorem 2.34 (Dragomir, 2010, [14]). Let J be an interval and g W J ! R be a
log-convex differentiable function on J̊ whose derivative g 0 is continuous on J̊. If A
is a selfadjoint operators on the Hilbert space H with Sp.A/ Œm; M J̊; then
.1 /
D
E
Am1H
M1H A
Œg.M / M m Œg.m/ M m x; x
hg.A/x; xi
h
D
0
g .M /
H/
g.A/ exp .M1H A/.Am1
M m
g.M /
hg.A/x; xi
g 0 .M / g 0 .m/
1
.M m/
exp
4
g.M /
g.m/
for each x 2 H with kxk D 1:
g 0 .m/
g.m/
i
x; x
E
(2.81)
2.6 Some Jensen’s Type Inequalities for Log-Convex Functions
41
Proof. Utilizing inequality (2.76) we have successively
and
g 0 .s/
g ..1 / t C s/
exp .1 /
.t s/
g.s/
g.s/
(2.82)
g ..1 / t C s/
g 0 .t/
exp .t s/
g.t/
g.t/
(2.83)
for any t; s 2J̊ and any 2 Œ0; 1 :
Now, if we take the power in inequality (2.82) and the power 1 in (2.83)
and multiply the obtained inequalities, we deduce
Œg.t/1 Œg.s/
g ..1 / t C s/
0
g .t/ g 0 .s/
.t s/
exp .1 / g.t/
g.s/
for any t; s 2J̊ and any 2 Œ0; 1 :
Further on, if we choose in (2.84) t D M; s D m and D
(2.84) we get the inequality
(2.84)
M u
;
M m
then, from
M u
um
Œg.M / M m Œg.m/ M m
g .u/
.M u/ .u m/ g 0 .M / g 0 .m/
exp
M m
g.M /
g.m/
(2.85)
which, together with the inequality
.M u/ .u m/
1
.M m/
M m
4
produce
um
M u
Œg.M / M m Œg.m/ M m
.M u/ .u m/ g 0 .M / g 0 .m/
g .u/ exp
M m
g.M /
g.m/
0
0
g .M / g .m/
1
.M m/
g .u/ exp
4
g.M /
g.m/
(2.86)
for any u 2 Œm; M :
If we apply property (P) to inequality (2.86) and for the operator A we deduce
the desired result.
t
u
42
2 Inequalities of the Jensen Type
2.6.3 More Inequalities for Differentiable Log-Convex Functions
The following results providing companion inequalities for the Jensen inequality for
differentiable log-convex functions obtained above hold:
Theorem 2.35 (Dragomir, 2010, [15]). Let A be a selfadjoint operator on the
Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with
m < M: If g W J ! .0; 1/ is a differentiable log-convex function with the
derivative continuous on JV and Œm; M JV , then
hg0 .A/Ax; xi hg.A/Ax; xi hg 0 .A/x; xi
hg.A/x; xi
hg.A/x; xi hg.A/x; xi
h
i
ln g.A/x;xi
exp hg.A/
hg.A/x;xi
1
g hg.A/Ax;xi
hg.A/x;xi
exp
for each x 2 H with kxk D 1:
If
hg 0 .A/Ax; xi
2 JV for each x 2 H with kxk D 1;
hg 0 .A/x; xi
(2.87)
(C)
then
2
exp 4
g0
g
hg 0 .A/Ax;xi
hg 0 .A/x;xi
hg 0 .A/Ax;xi
hg 0 .A/x;xi
3
hg 0 .A/Ax; xi hAg.A/x; xi 5
hg 0 .A/x; xi
hg.A/x; xi
0
.A/Ax;xi
g hg
0
hg .A/x;xi
1
hg.A/ ln g.A/x;xi
exp
hg.A/x;xi
(2.88)
for each x 2 H with kxk D 1:
Proof. By the gradient inequality for the convex function ln g we have
g 0 .s/
g 0 .t/
.t s/ ln g.t/ ln g.s/ .t s/
g.t/
g.s/
(2.89)
for any t; s 2 JV , which by multiplication with g.t/ > 0 is equivalent with
g 0 .t/.t s/ g.t/ ln g.t/ g.t/ ln g.s/ for any t; s 2 JV :
g 0 .s/
.tg.t/ sg.t//
g.s/
(2.90)
2.6 Some Jensen’s Type Inequalities for Log-Convex Functions
43
Fix s 2 JV and apply property (P) to get that
˝
˛
˝
˛
g 0 .A/Ax; x s g 0 .A/x; x
hg.A/ ln g.A/x; xi hg.A/x; xi ln g.s/
g 0 .s/
.hAg.A/x; xi s hg.A/x; xi/
g.s/
(2.91)
for any x 2 H with kxk D 1; which is an inequality of interest in itself as well.
Since
hg.A/Ax; xi
2 Œm; M for any x 2 H with kxk D 1
hg.A/x; xi
then on choosing s WD
hg.A/Ax;xi
hg.A/x;xi
in (2.91) we get
˝ 0
˛ hg.A/Ax; xi ˝ 0
˛
g .A/Ax; x g .A/x; x
hg.A/x; xi
hg.A/ ln g.A/x; xi hg.A/x; xi ln g
hg.A/Ax; xi
hg.A/x; xi
0;
which, by division with hg.A/x; xi > 0; produces
hg 0 .A/Ax; xi hg.A/Ax; xi hg 0 .A/x; xi
hg.A/x; xi
hg.A/x; xi hg.A/x; xi
hg.A/ ln g.A/x; xi
hg.A/Ax; xi
ln g
0
hg.A/x; xi
hg.A/x; xi
(2.92)
for any x 2 H with kxk D 1:
Taking the exponential in (2.92) we deduce the desired inequality (2.87).
0 .A/Ax;xi
Now, assuming that condition (C) holds, then by choosing s WD hg
hg 0 .A/x;xi in
(2.91) we get
0
hg .A/Ax; xi
0 hg.A/ ln g.A/x; xi hg.A/x; xi ln g
hg 0 .A/x; xi
0
.A/Ax;xi g 0 hg
hg 0 .A/x;xi
hg 0 .A/Ax; xi
hAg.A/x; xi 0
xi
hg.A/x;
hg 0 .A/x; xi
g hg .A/Ax;xi
hg 0 .A/x;xi
which, by dividing with hg.A/x; xi > 0 and rearranging, is equivalent with
g0
g
hg 0 .A/Ax;xi
hg 0 .A/x;xi
hg 0 .A/Ax;xi
hg 0 .A/x;xi
hg 0 .A/Ax; xi hAg.A/x; xi
hg 0 .A/x; xi
hg.A/x; xi
44
2 Inequalities of the Jensen Type
hg 0 .A/Ax; xi
ln g
hg 0 .A/x; xi
hg.A/ ln g.A/x; xi
0
hg.A/x; xi
(2.93)
for any x 2 H with kxk D 1:
Finally, on taking the exponential in (2.93) we deduce the desired inequality
(2.88).
t
u
Remark 2.36. We observe that a sufficient condition for (C) to hold is that either
g 0 .A/ or g 0 .A/ is a positive definite operator on H:
Corollary 2.37 (Dragomir, 2010, [15]). Assume that A and g are as in Theorem
2.35. If condition (C) holds, then we have the double inequality
0
hg .A/Ax; xi
hg.A/ ln g.A/x; xi
ln g
hg 0 .A/x; xi
hg.A/x; xi
hg.A/Ax; xi
(2.94)
ln g
hg.A/x; xi
for any x 2 H with kxk D 1:
The following result providing different inequalities also holds:
Theorem 2.38 (Dragomir, 2010, [15]). Let A be a selfadjoint operator on the
Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with
m < M: If g W J ! .0; 1/ is a differentiable log-convex function with the
derivative continuous on JV and Œm; M JV , then
hg.A/Ax; xi
exp g0 .A/ A x; x
1H
hg.A/x; xi
0
1g.A/
*
+
g.A/
@
A
x; x
g hg.A/Ax;xi
hg.A/x;xi
2 hg.A/Ax;xi 3
*
+
g 0 hg.A/x;xi xi
hg.A/Ax;
Ag.A/ exp 4 g.A/ 5 x; x 1 (2.95)
hg.A/x; xi
g hg.A/Ax;xi
hg.A/x;xi
for each x 2 H with kxk D 1:
If condition (C) from Theorem 2.35 holds, then
2 hg0 .A/Ax;xi 3
*
+
g 0 hg0 .A/x;xi hg 0 .A/Ax; xi
exp 4 0
g.A/ Ag.A/ 5 x; x
hg 0 .A/x; xi
g hg .A/Ax;xi
hg 0 .A/x;xi
* +
g.A/
hg 0 .A/Ax; xi
1
Œg.A/
g
x; x
hg 0 .A/x; xi
2.6 Some Jensen’s Type Inequalities for Log-Convex Functions
hg 0 .A/Ax; xi
1H A x; x 1
exp g .A/
hg 0 .A/x; xi
0
45
(2.96)
for each x 2 H with kxk D 1:
Proof. By taking the exponential in (2.90) we have the following inequality:
0
0
g.t/ g.t /
g .s/
.tg.t/ sg.t//
(2.97)
exp
exp g .t/.t s/ g.s/
g.s/
for any t; s 2 JV :
If we fix s 2 JV and apply property (P) to inequality (2.97), we deduce
*
+
˝
0
˛
g.A/ g.A/
exp g .A/ .A s1H / x; x x; x
g.s/
0
g .s/
.Ag.A/ sg.A// x; x
exp
g.s/
(2.98)
for each x 2 H with kxk D 1; where 1H is the identity operator on H:
By Mond–Pečarić’s inequality applied for the convex function exp, we also have
0
g .s/
.Ag.A/ sg.A// x; x
exp
g.s/
0
g .s/
exp
.hAg.A/x; xi s hg.A/x; xi/
(2.99)
g.s/
for each s 2 JV and x 2 H with kxk D 1:
Now, if we choose s WD hg.A/Ax;xi
2 Œm; M in (2.98) and (2.99) we deduce the
hg.A/x;xi
desired result (2.95).
Observe that, inequality (2.97) is equivalent with
0
g.s/ g.t /
g .s/
.sg.t/ tg.t// exp
exp g 0 .t/ .s t/
(2.100)
g.s/
g.t/
for any t; s 2 JV :
If we fix s 2 JV and apply property (P) to inequality (2.100) we deduce
0
g .s/
exp
.sg.A/ Ag.A// x; x
g.s/
g.A/
1
g.s/ Œg.A/
x; x
˝
˛
exp g 0 .A/ .s1H A/ x; x
for each x 2 H with kxk D 1:
(2.101)
46
2 Inequalities of the Jensen Type
By Mond–Pečarić’s inequality we also have
˝
˛
˝
˛ ˝
˛
exp g 0 .A/ .s1H A/ x; x exp s g 0 .A/x; x g 0 .A/Ax; x
(2.102)
for each s 2 JV and x 2 H with kxk D 1:
Taking into account that condition (C) is valid, then we can choose in (2.101)
0 .A/Ax;xi
and (2.102) s WD hg
t
u
hg 0 .A/x;xi to get the desired result (2.96).
2.6.4 A Reverse Inequality
The following reverse inequality is also of interest:
Theorem 2.39 (Dragomir, 2010, [15]). Let A be a selfadjoint operator on the
Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with
m < M: If g W J ! .0; 1/ is a differentiable log-convex function with the
derivative continuous on JV and Œm; M JV , then
M hAx;xi
hAx;xim
Œg.m/ M m Œg.M / M m
.1 /
exp hln g.A/x; xi
h.M1H A/ .A m1H / x; xi g 0 .M / g 0 .m/
exp
M m
g.M /
g.m/
0
0
.M hAx; xi/ .hAx; xi m/ g .M / g .m/
exp
M m
g.M /
g.m/
0
1
g .M / g 0 .m/
exp
(2.103)
.M m/
4
g.M /
g.m/
for each x 2 H with kxk D 1:
Proof. Utilizing inequality (2.89) we have successively
ln g ..1 / t C s/ ln g.s/ .1 /
g 0 .s/
.t s/
g.s/
and
ln g ..1 / t C s/ ln g.t/ g 0 .t/
.t s/
g.t/
(2.104)
(2.105)
for any t; s 2J̊ and any 2 Œ0; 1 :
Now, if we multiply (2.104) by and (2.105) by 1 and sum the obtained
inequalities, we deduce
2.7 Hermite–Hadamard’s Type Inequalities
47
.1 / ln g.t/ C ln g.s/ ln g ..1 / t C s/
0
g .t/ g 0 .s/
.t s/
.1 / g.t/
g.s/
(2.106)
for any t; s 2J̊ and any 2 Œ0; 1 :
M u
Now, if we choose WD M
; s WD m and t WD M in (2.106) then we get the
m
inequality
M u
um
ln g.M / C
ln g.m/ ln g .u/
M m
M m
.M u/ .u m/ g 0 .M / g 0 .m/
M m
g.M /
g.m/
(2.107)
for any u 2 Œm; M :
If we use property (P) for the operator A we get
M hAx; xi
hAx; xi m
ln g.M / C
ln g.m/ hln g.A/x; xi
M m
M m
h.M1H A/ .A m1H / x; xi g 0 .M / g 0 .m/
M m
g.M /
g.m/
(2.108)
for each x 2 H with kxk D 1:
Taking the exponential in (2.108) we deduce the first inequality in (2.103).
Now, consider the function h W Œm; M ! R, h.t/ D .M t/ .t m/ : This
function is concave in Œm; M and by Mond–Pečarić’s inequality we have
h.M1H A/ .A m1H / x; xi .M hAx; xi/ .hAx; xi m/
for each x 2 H with kxk D 1; which proves the second inequality in (2.103).
For the last inequality, we observe that
.M hAx; xi/ .hAx; xi m/ 1
.M m/2 ;
4
and the proof is complete.
t
u
2.7 Hermite–Hadamard’s Type Inequalities
2.7.1 Scalar Case
If f W I ! R is a convex function on the interval I; then for any a; b 2 I with
a ¤ b we have the following double inequality:
48
2 Inequalities of the Jensen Type
f
aCb
2
1
ba
Z
a
b
f .t/dt f .a/ C f .b/
:
2
(HH)
This remarkable result is well known in the literature as the Hermite–Hadamard
inequality [24].
For various generalizations, extensions, reverses and related inequalities, see
[1, 2, 18, 20–24] the monograph [17] and the references therein.
2.7.2 Some Inequalities for Convex Functions
The following inequality related to the Mond–Pečarić result also holds:
Theorem 2.40 (Dragomir, 2010, [13]). Let A be a selfadjoint operator on the
Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with
m < M:
If f is a convex function on Œm; M ; then
f .m/ C f .M /
f .A/ C f ..m C M / 1H A/
x; x
2
2
f .hAx; xi/ C f .m C M hAx; xi/
2
mCM
f
2
for each x 2 H with kxk D 1:
In addition, if x 2 H with kxk D 1 and hAx; xi ¤
mCM
2 ;
(2.109)
then also
f .hAx; xi/ C f .m C M hAx; xi/
2
Z mCM hAx;xi
mCM
2
:
f .u/ du f
mCM
2
hAx; xi hAx;xi
2
(2.110)
Proof. Since f is convex on Œm; M then for each u 2 Œm; M we have the
inequalities
M u
um
f .m/ C
f .M /
M m
M m
um
M u
mC
M D f .u/
f
M m
M m
(2.111)
2.7 Hermite–Hadamard’s Type Inequalities
49
and
um
M u
f .M / C
f .m/ f
M m
M m
M u
um
MC
m
M m
M m
D f .M C m u/ :
(2.112)
If we add these two inequalities we get
f .m/ C f .M / f .u/ C f .M C m u/
for any u 2 Œm; M ; which, by property (P) applied for the operator A; produces the
first inequality in (2.109).
By the Mond–Pečarić inequality we have
hf ..m C M / 1H A/ x; xi f .m C M hAx; xi/;
which together with the same inequality produces the second inequality in (2.109).
The third part follows by the convexity of f:
In order to prove (2.110), we use the Hermite–Hadamard inequality (HH) for the
convex functions f and the choices a D hAx; xi and b D m C M hAx; xi:
The proof is complete.
t
u
Remark 2.41. We observe that, from inequality (2.109) we have the following
inequality in the operator order of B .H /:
f .m/ C f .M /
f .A/ C f ..m C M / 1H A/
1H 2
2
mCM
1H ;
f
2
(2.113)
where f is a convex function on Œm; M and A a selfadjoint operator on the Hilbert
space H with Sp.A/ Œm; M for some scalars m; M with m < M:
The case of log-convex functions may be of interest for applications and therefore
is stated in:
Corollary 2.42 (Dragomir, 2010, [13]). If g is a log-convex function on Œm; M ;
then
D
E
p
g.m/g.M / exp ln Œg.A/g ..m C M / 1H A/1=2 x; x
p
g .hAx; xi/ g .m C M hAx; xi/
mCM
(2.114)
g
2
for each x 2 H with kxk D 1:
50
2 Inequalities of the Jensen Type
In addition, if x 2 H with kxk D 1 and hAx; xi ¤
p
mCM
2 ;
then also
g .hAx; xi/ g .m C M hAx; xi/
"
#
Z mCM hAx;xi
2
exp mCM
ln g .u/ du
hAx; xi hAx;xi
2
mCM
g
:
2
(2.115)
The following result also holds:
Theorem 2.43 (Dragomir, 2010, [13]). Let A and B selfadjoint operators on the
Hilbert space H and assume that Sp.A/; Sp.B/ Œm; M for some scalars m; M
with m < M:
If f is a convex function on Œm; M ; then
f
ACB
x; x
2
1
Œf ..1 t/ hAx; xi C t hBx; xi/ C f .t hAx; xi C .1 t/ hBx; xi/
2
1
Œf ..1 t/ A C tB/ C f .tA C .1 t/ B/ x; x
2
˝ ACB
˝
˛
˛
M ACB
x; x
x; x m
2
2
f .m/ C
f .M /
(2.116)
M m
M m
for any t 2 Œ0; 1 and each x 2 H with kxk D 1:
Moreover, we have the Hermite–Hadamard’s type inequalities:
ACB
f
x; x
2
Z 1
f ..1 t/ hAx; xi C t hBx; xi/ dt
0
Z
1
f ..1 t/ A C tB/ dt x; x
0
˝ ACB
˝
˛
˛
M ACB
x; x
x; x m
2
2
f .m/ C
f .M /
M m
M m
each x 2 H with kxk D 1:
(2.117)
2.7 Hermite–Hadamard’s Type Inequalities
51
In addition, if we assume that B A is a positive definite operator, then
ACB
f
x; x h.B A/ x; xi
2
Z
Z hBx;xi
f .u/ du h.B A/ x; xi
hAx;xi
"
˝ ACB
˛
1
f ..1 t/ A C tB/ dt x; x
0
M 2 x; x
f .m/ C
h.B A/ x; xi
M m
#
˛
x;
x
m
2
f .M / : (2.118)
M m
˝ ACB
Proof. It is obvious that for any t 2 Œ0; 1 we have
Sp ..1 t/ A C tB/; Sp .tA C .1 t/ B/ Œm; M :
On making use of the Mond–Pečarić inequality we have
f ..1 t/ hAx; xi C t hBx; xi/ hf ..1 t/ A C tB/ x; xi
(2.119)
f .t hAx; xi C .1 t/ hBx; xi/ hf .tA C .1 t/ B/ x; xi
(2.120)
and
for any t 2 Œ0; 1 and each x 2 H with kxk D 1:
Adding (2.119) with (2.120) and utilizing the convexity of f we deduce the first
two inequalities in (2.116).
By the Lah–Ribarić inequality (2.69) we also have
hf ..1 t/ A C tB/ x; xi M .1 t/ hAx; xi t hBx; xi
f .m/
M m
.1 t/ hAx; xi C t hBx; xi m
f .M / (2.121)
C
M m
and
hf .tA C .1 t/ B/ x; xi M t hAx; xi .1 t/ hBx; xi
f .m/
M m
t hAx; xi C .1 t/ hBx; xi m
C
f .M / (2.122)
M m
for any t 2 Œ0; 1 and each x 2 H with kxk D 1:
Now, if we add inequalities (2.121) with (2.122) and divide by two, we deduce
the last part in (2.116).
52
2 Inequalities of the Jensen Type
Integrating the inequality over t 2 Œ0; 1, utilizing the continuity property of the
inner product and the properties of the integral of operator-valued functions we have
f
ACB
x; x
2
1
2
Z
1
f ..1 t/ hAx; xi C t hBx; xi/ dt
0
Z
1
C
f .t hAx; xi C .1 t/ hBx; xi/ dt
0
Z 1
1
f ..1 t/ A C tB/ dt
2 0
Z 1
f .tA C .1 t/ B/ dt x; x
C
0
˝ ACB
˛
˛
M 2 x; x
2 x; x m
f .m/ C
f .M /: (2.123)
M m
M m
˝ ACB
Since
Z
Z
1
1
f ..1 t/ hAx; xi C t hBx; xi/ dt D
f .t hAx; xi C .1 t/ hBx; xi/ dt
0
0
and
Z
Z
1
f ..1 t/ A C tB/ dt D
0
1
f .tA C .1 t/ B/ dt
0
then, by (2.123), we deduce inequality (2.117).
Inequality (2.118) follows from (2.117) by observing that for hBx; xi > hAx; xi
we have
Z
1
0
1
f ..1 t/ hAx; xi C t hBx; xi/ dt D
hBx; xi hAx; xi
Z
hBx;xi
f .u/ du
hAx;xi
for each x 2 H with kxk D 1:
Remark 2.44. We observe that, from inequalities (2.116) and (2.117) we have the
following inequalities in the operator order of B .H /:
1
Œf ..1 t/ A C tB/ C f .tA C .1 t/ B/
2
f .m/
ACB
M1H ACB
m1H
2
C f .M / 2
;
M m
M m
(2.124)
where f is a convex function on Œm; M and A; B are selfadjoint operator on the
Hilbert space H with Sp.A/; Sp.B/ Œm; M for some scalars m; M with m < M:
2.7 Hermite–Hadamard’s Type Inequalities
53
The case of log-convex functions is as follows:
Corollary 2.45 (Dragomir, 2010, [13]). If g is a log-convex function on Œm; M ;
then
ACB
x; x
g
2
p
g ..1 t/ hAx; xi C t hBx; xi/ g .t hAx; xi C .1 t/ hBx; xi/
1
exp Œln g ..1 t/ A C tB/ C ln g .tA C .1 t/ B/ x; x
2
g.m/
h ACB
2 x;x i
M
M m
g.M /
h ACB
2 x;x im
(2.125)
M m
for any t 2 Œ0; 1 and each x 2 H with kxk D 1:
Moreover, we have the Hermite–Hadamard’s type inequalities:
ACB
x; x
2
Z 1
exp
ln g ..1 t/ hAx; xi C t hBx; xi/ dt
g
0
Z
1
exp
ln g ..1 t/ A C tB/ dt x; x
0
g.m/
h ACB
2 x;x i
M
M m
g.M /
h ACB
2 x;x im
M m
(2.126)
for each x 2 H with kxk D 1:
In addition, if we assume that B A is a positive definite operator, then
h.BA/x;xi
ACB
g
x; x
2
"Z
#
hBx;xi
exp
ln g .u/ du
hAx;xi
Z
1
exp h.B A/ x; xi
"
g.m/
ln g ..1 t/ A C tB/ dt x; x
0
h ACB
2 x;x i
M
M m
for each x 2 H with kxk D 1:
g.M /
h ACB
2 x;x im
M m
#h.BA/x;xi
(2.127)
54
2 Inequalities of the Jensen Type
From a different perspective, we have the following result as well:
Theorem 2.46 (Dragomir, 2010, [13]). Let A and B selfadjoint operators on the
Hilbert space H and assume that Sp.A/; Sp.B/ Œm; M for some scalars m; M
with m < M: If f is a convex function on Œm; M ; then
hAx; xi C hBy; yi
2
Z 1
f ..1 t/ hAx; xi C t hBy; yi/ dt
f
0
Z
1
0
f ..1 t/ A C t hBy; yi 1H / dt x; x
1
Œhf .A/x; xi C f .hBy; yi/
2
1
Œhf .A/x; xi C hf .B/y; yi
2
(2.128)
and
f
hAx; xi C hBy; yi
2
A C hBy; yi 1H
f
x; x
2
Z 1
f ..1 t/ A C t hBy; yi 1H / dt x; x (2.129)
0
for each x; y 2 H with kxk D kyk D 1:
Proof. For a convex function f and any u; v 2 Œm; M and t 2 Œ0; 1, we have the
double inequality:
f
uCv
2
1
Œf ..1 t/ u C tv/ C f .tu C .1 t/ v/
2
1
Œf .u/ C f .v/ :
2
(2.130)
Utilizing the second inequality in (2.130) we have
1
f ..1 t/ u C t hBy; yi/ C f .tu C .1 t/ hBy; yi/
2
1
Œf .u/ C f .hBy; yi/
2
for any u 2 Œm; M , t 2 Œ0; 1 and y 2 H with kyk D 1:
(2.131)
2.7 Hermite–Hadamard’s Type Inequalities
55
Now, on applying property (P) to inequality (2.131) for the operator A we have
1
Œhf ..1 t/ A C t hBy; yi/ x; xi C hf .tA C .1 t/ hBy; yi/ x; xi
2
1
Œhf .A/x; xi C f .hBy; yi/
(2.132)
2
for any t 2 Œ0; 1 and x; y 2 H with kxk D kyk D 1:
On applying the Mond–Pečarić inequality we also have
1
f ..1 t/ hAx; xi C t hBy; yi/ C f .t hAx; xi C .1 t/ hBy; yi/
2
1
Œhf ..1 t/ A C t hBy; yi 1H / x; xi C hf .tA C .1 t/ hBy; yi 1H / x; xi
2
(2.133)
for any t 2 Œ0; 1 and x; y 2 H with kxk D kyk D 1:
Now, integrating over t on Œ0; 1 inequalities (2.132) and (2.133) and taking into
account that
Z 1
hf ..1 t/ A C t hBy; yi 1H / x; xi dt
0
Z
1
D
hf .tA C .1 t/ hBy; yi 1H / x; xi dt
0
Z
1
D
0
f ..1 t/ A C t hBy; yi 1H / dt x; x
and
Z
Z
1
1
f ..1 t/ hAx; xi C t hBy; yi/ dt D
0
f .t hAx; xi C .1 t/ hBy; yi/ dt;
0
we obtain the second and the third inequality in (2.128).
Further, on applying the Jensen integral inequality for the convex function f we
also have
Z 1
f ..1 t/ hAx; xi C t hBy; yi/ dt
0
Z
1
f
Df
Œ.1 t/ hAx; xi C t hBy; yi dt
0
hAx; xi C hBy; yi
2
for each x; y 2 H with kxk D kyk D 1, proving the first part of (2.128).
56
2 Inequalities of the Jensen Type
Now, on utilizing the first part of (2.130) we can also state that
f
u C hBy; yi
2
1
Œf ..1 t/ u C t hBy; yi/ C f .tu C .1 t/ hBy; yi/
2
(2.134)
for any u 2 Œm; M , t 2 Œ0; 1 and y 2 H with kyk D 1:
Further, on applying property (P) to inequality (2.134) and for the operator A
we get
A C hBy; yi 1H
f
x; x
2
1
Œhf ..1 t/ A C t hBy; yi 1H / x; xi C hf .tA C .1 t/ hBy; yi 1H / x; xi
2
for each x; y 2 H with kxk D kyk D 1; which, by integration over t in Œ0; 1
produces the second inequality in (2.129). The first inequality is obvious.
u
t
Remark 2.47. It is important to remark that, from inequalities (2.128) and (2.129)
we have the following Hermite–Hadamard’s type results in the operator order of
B .H / and for the convex function f W Œm; M ! R:
Z 1
A C hBy; yi 1H
f
f ..1 t/ A C t hBy; yi 1H / dt
2
0
1
Œf .A/ C f .hBy; yi/ 1H 2
(2.135)
for any y 2 H with kyk D 1 and any selfadjoint operators A; B with spectra in
Œm; M :
In particular, we have from (2.135)
f
A C hAy; yi 1H
2
Z
1
0
f ..1 t/ A C t hAy; yi 1H / dt
1
Œf .A/ C f .hAy; yi/ 1H 2
(2.136)
for any y 2 H with kyk D 1 and
Z 1
A C s1H
f
f ..1 t/ A C ts1H / dt
2
0
for any s 2 Œm; M :
1
Œf .A/ C f .s/1H 2
(2.137)
2.7 Hermite–Hadamard’s Type Inequalities
57
As a particular case of the above theorem, we have the following refinement of
the Mond–Pečarić inequality:
Corollary 2.48 (Dragomir, 2010, [13]). Let A be a selfadjoint operator on the
Hilbert space H and assume that Sp.A/ Œm; M for some scalars m; M with
m < M: If f is a convex function on Œm; M ; then
A C hAx; xi 1H
x; x
f .hAx; xi/ f
2
Z 1
f ..1 t/ A C t hAx; xi 1H / dt x; x
0
1
Œhf .A/x; xi C f .hAx; xi/ hf .A/x; xi :
2
(2.138)
Finally, the case of log-convex functions is as follows:
Corollary 2.49 (Dragomir, 2010, [13]). If g is a log-convex function on Œm; M ;
then
hAx; xi C hBy; yi
g
2
Z 1
exp
ln g ..1 t/ hAx; xi C t hBy; yi/ dt
0
Z
1
exp
0
ln g ..1 t/ A C t hBy; yi 1H / dt x; x
1
Œhln g.A/x; xi C ln g .hBy; yi/
exp
2
1
exp
Œhln g.A/x; xi C hln g.B/y; yi
2
(2.139)
and
hAx; xi C hBy; yi
g
2
A C hBy; yi 1H
exp ln g
x; x
2
Z 1
exp
ln g ..1 t/ A C t hBy; yi 1H / dt x; x
0
(2.140)
58
2 Inequalities of the Jensen Type
and
A C hAx; xi 1H
x; x
g .hAx; xi/ exp ln g
2
Z 1
exp
ln g ..1 t/ A C t hAx; xi 1H / dt x; x
exp
0
1
Œhln g.A/x; xi C ln g .hAx; xi/
2
exp hln g.A/x; xi
(2.141)
respectively, for each x 2 H with kxk D 1 and A; B selfadjoint operators with
spectra in Œm; M :
It is obvious that all the above inequalities can be applied for particular convex
or log-convex functions of interest. The details are left to the interested reader.
2.8 Hermite–Hadamard’s Type Inequalities for Operator
Convex Functions
2.8.1 Introduction
The following inequality holds for any convex function f defined on R:
.b a/f
aCb
2
Z
<
b
f .x/dx < .b a/
a
f .a/ C f .b/
;
2
a; b 2 R: (2.142)
It was first discovered by Ch. Hermite in 1881 in the journal Mathesis (see [24]).
But this result was nowhere mentioned in the mathematical literature and was not
widely known as Hermite’s result [27].
E.F. Beckenbach, a leading expert on the history and the theory of convex
functions, wrote that this inequality was proven by J. Hadamard in 1893 [3]. In
1974, D.S. Mitrinović found Hermite’s note in Mathesis [24]. Since (2.142) was
known as Hadamard’s inequality, the inequality is now commonly referred as the
Hermite–Hadamard inequality [27].
Let X be a vector space, x; y 2 X; x ¤ y. Define the segment
Œx; y WD f.1 t/x C ty; t 2 Œ0; 1g:
2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions
59
We consider the function f W Œx; y ! R and the associated function
g.x; y/ W Œ0; 1 ! R; g.x; y/.t/ WD f Œ.1 t/x C ty; t 2 Œ0; 1:
Note that f is convex on Œx; y if and only if g.x; y/ is convex on Œ0; 1.
For any convex function defined on a segment Œx:y X , we have the Hermite–
Hadamard integral inequality (see [4, p. 2])
f
xCy
2
Z
1
0
f Œ.1 t/x C tydt f .x/ C f .y/
;
2
(2.143)
which can be derived from the classical Hermite–Hadamard inequality (2.142) for
the convex function g.x; y/ W Œ0; 1 ! R.
Since f .x/ D kxkp (x 2 X and 1 p < 1) is a convex function, we have the
following norm inequality from (2.143) (see [26, p. 106]):
Z 1
x C y p
kxkp C kykp
p
k.1
t/x
C
tyk
dt
2 2
0
(2.144)
for any x; y 2 X .
Motivated by the above results, we investigate in this section the operator version
of the Hermite–Hadamard inequality for operator convex functions. The operator
quasi-linearity of some associated functionals are also provided.
A real-valued continuous function f on an interval I is said to be operator
convex (operator concave) if
f ..1 / A C B/ ./ .1 / f .A/ C f .B/
(OC)
in the operator order, for all 2 Œ0; 1 and for every selfadjoint operator A and B
on a Hilbert space H whose spectra are contained in I: Notice that a function f is
operator concave if f is operator convex.
A real-valued continuous function f on an interval I is said to be operator
monotone if it is monotone with respect to the operator order, i.e. A B with
Sp.A/; Sp.B/ I imply f .A/ f .B/:
For some fundamental results on operator convex (operator concave) and operator monotone functions, see [19] and the references therein.
As examples of such functions, we note that f .t/ D t r is operator monotone on
Œ0; 1/ if and only if 0 r 1: The function f .t/ D t r is operator convex on
.0; 1/ if either 1 r 2 or 1 r 0 and is operator concave on .0; 1/ if
0 r 1: The logarithmic function f .t/ D ln t is operator monotone and operator
concave on .0; 1/: The entropy function f .t/ D t ln t is operator concave on
.0; 1/: The exponential function f .t/ D et is neither operator convex nor operator
monotone.
60
2 Inequalities of the Jensen Type
2.8.2 Some Hermite–Hadamard’s Type Inequalities
We start with the following result:
Theorem 2.50 (Dragomir, 2010, [12]). Let f W I ! R be an operator convex
function on the interval I: Then for any selfadjoint operators A and B with spectra
in I we have the inequality
1
3A C B
A C 3B
ACB
f
Cf
f
2
2
4
4
Z 1
f ..1 t/A C tB/dt
0
ACB
f .A/ C f .B/
f .A/ C f .B/
1
f
C
:
2
2
2
2
(2.145)
Proof. First of all, since the function f is continuous, the operator-valued integral
R1
0 f ..1 t/ACtB/dt exists for any selfadjoint operators A and B with spectra in I:
We give here two proofs, the first using only the definition of operator convex
functions and the second using the classical Hermite–Hadamard inequality for realvalued functions.
1. By the definition of operator convex functions we have the double inequality:
f
C CD
2
1
Œf ..1 t/ C C tD/ C f ..1 t/ D C tC /
2
1
Œf .C / C f .D/
2
(2.146)
for any t 2 Œ0; 1 and any selfadjoint operators C and D with the spectra in I:
Integrating inequality (2.146) over t 2 Œ0; 1 and taking into account that
Z
1
Z
1
f ..1 t/ C C tD/ dt D
0
f ..1 t/ D C tC / dt
0
then we deduce the Hermite–Hadamard inequality for operator convex functions
f
C CD
2
Z
1
f ..1 t/ C C tD/ dt 0
1
Œf .C / C f .D/
2
that holds for any selfadjoint operators C and D with the spectra in I:
(HHO)
2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions
61
Now, on making use of the change of variable u D 2t we have
Z
1=2
f ..1 t/A C tB/dt D
0
1
2
Z
1
f
0
ACB
du
.1 u/ A C u
2
and by the change of variable u D 2t 1 we have
Z
1
1
f ..1 t/A C tB/dt D
2
1=2
Z
1
0
ACB
C uB du:
f .1 u/
2
Utilizing the Hermite–Hadamard inequality (HHO) we can write
f
3A C B
4
ACB
du
.1 u/ A C u
2
0
1
ACB
f .A/ C f
2
2
Z
1
f
and
f
A C 3B
4
ACB
C uB du
.1 u/
2
0
1
ACB
f .A/ C f
;
2
2
Z
1
f
which by summation and division by two produces the desired result (2.145).
2. Consider now x 2 H; kxk D 1 and two selfadjoint operators A and B with
spectra in I . Define the real-valued function ' x;A;B W Œ0; 1 ! R given by
' x;A;B .t/ D hf ..1 t/A C tB/x; xi:
Since f is operator convex, then for any t1 ; t2 2 Œ0; 1 and ˛; ˇ 0 with
˛ C ˇ D 1 we have
' x;A;B .˛t1 C ˇt2 /
D hf ..1 .˛t1 C ˇt2 // A C .˛t1 C ˇt2 / B/ x; xi
D hf .˛ Œ.1 t1 / A C t1 B C ˇ Œ.1 t2 / A C t2 B/ x; xi
˛ hf .Œ.1 t1 / A C t1 B/ x; xi C ˇ hf .Œ.1 t2 / A C t2 B/ x; xi
D ˛' x;A;B .t1 / C ˇ' x;A;B .t2 /
showing that ' x;A;B is a convex function on Œ0; 1 :
62
2 Inequalities of the Jensen Type
Now we use the Hermite–Hadamard inequality for real-valued convex
functions
Z b
1
aCb
g.a/ C g.b/
g
g.s/ds 2
ba a
2
to get that
' x;A;B
and
Z 1=2
' x;A;B .0/ C ' x;A;B 12
1
2
' x;A;B .t/dt 4
2
0
Z 1
' x;A;B 12 C ' x;A;B .1/
3
2
' x;A;B .t/dt ' x;A;B
4
2
1=2
which by summation and division by two produces
1
2
3A C B
A C 3B
f
Cf
x; x
4
4
Z 1
hf ..1 t/A C tB/x; xi dt
0
1
2
ACB
f .A/ C f .B/
f
C
x; x :
2
2
(2.147)
Finally, since by the continuity of the function f we have
Z
Z
1
0
1
f ..1 t/A C tB/dtx; x
hf ..1 t/A C tB/x; xi dt D
0
for any x 2 H; kxk D 1 and any two selfadjoint operators A and B with spectra
in I; we deduce from (2.147) the desired result (2.145).
t
u
A simple consequence of the above theorem is that the integral is closer to the
left bound than to the right, namely we can state:
Corollary 2.51 (Dragomir, 2010, [12]). With the assumptions in Theorem 2.50 we
have the inequality
Z
1
f ..1 t/A C tB/dt f
.0 /
0
f .A/ C f .B/
2
Z
ACB
2
1
f ..1 t/A C tB/dt:
(2.148)
0
Remark 2.52. Utilizing different examples of operator convex or concave functions,
we can provide inequalities of interest.
2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions
63
If r 2 Œ1; 0 [ Œ1; 2 then we have the inequalities for powers of operators
ACB r
3A C B r
1
A C 3B r
C
2
2
4
4
Z 1
..1 t/A C tB/r dt
0
1
2
ACB
2
r
C
Ar C B r
2
Ar C B r
2
(2.149)
for any two selfadjoint operators A and B with spectra in .0; 1/ :
If r 2 .0; 1/ the inequalities in (2.149) hold with “ ” instead of “ ”:
We also have the following inequalities for logarithm:
ACB
1
3A C B
A C 3B
ln
ln
C ln
2
2
4
4
Z 1
ln..1 t/A C tB/dt
0
1
ACB
ln.A/ C ln.B/
ln.A/ C ln.B/
ln
C
2
2
2
2
(2.150)
for any two selfadjoint operators A and B with spectra in .0; 1/:
2.8.3 Some Operator Quasi-linearity Properties
Consider an operator convex function f W I R ! R defined on the
interval I and two distinct selfadjoint operators A; B with the spectra in I .
We denote by ŒA; B the closed operator segment defined by the family of operators
f.1 t/A C tB, t 2 Œ0; 1g : We also define the operator-valued functional
f .A; BI t/ WD .1 t/f .A/ C tf .B/ f ..1 t/A C tB/ 0
(2.151)
in the operator order, for any t 2 Œ0; 1 :
The following result concerning an operator quasi-linearity property for the
functional f .; I t/ may be stated:
Theorem 2.53 (Dragomir, 2010, [12]). Let f W I R ! R be an operator
convex function on the interval I . Then for each A; B two distinct selfadjoint
operators with the spectra in I and C 2 ŒA; B we have
.0 / f .A; C I t/ C f .C; BI t/ f .A; BI t/
(2.152)
64
2 Inequalities of the Jensen Type
for each t 2 Œ0; 1 ; i.e. the functional f .; I t/ is operator super-additive as a
function of interval.
If ŒC; D ŒA; B; then
.0 / f .C; DI t/ f .A; BI t/
(2.153)
for each t 2 Œ0; 1 ; i.e. the functional f .; I t/ is operator non-decreasing as a
function of interval.
Proof. Let C D .1 s/ A C sB with s 2 .0; 1/ : For t 2 .0; 1/ we have
f .C; BI t/ D .1 t/f ..1 s/ A C sB/ C tf .B/
f ..1 t/ Œ.1 s/ A C sB C tB/
and
f .A; C I t/ D .1 t/f .A/ C tf ..1 s/ A C sB/
f ..1 t/A C t Œ.1 s/ A C sB/
giving that
f .A; C I t/ C f .C; BI t/ f .A; BI t/
D f ..1 s/ A C sB/ C f ..1 t/A C tB/
f ..1 t/ .1 s/ A C Œ.1 t/s C t B/ f ..1 ts/ A C tsB/ : (2.154)
Now, for a convex function ' W I R ! R, where I is an interval, and any real
numbers t1 ; t2 ; s1 and s2 from I and with the properties that t1 s1 and t2 s2 we
have that
' .t1 / ' .t2 /
' .s1 / ' .s2 /
:
t1 t2
s1 s2
Indeed, since ' is convex on I then for any a 2 I the function
.t/ WD
(2.155)
W I n fag ! R
'.t/ '.a/
t a
is monotonic non-decreasing where is defined. Utilizing this property repeatedly we
have
' .t1 / ' .t2 /
' .s1 / ' .t2 /
' .t2 / ' .s1 /
D
t1 t2
s1 t2
t2 s1
which proves inequality (2.155).
' .s2 / ' .s1 /
' .s1 / ' .s2 /
D
;
s2 s1
s1 s2
2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions
65
For a vector x 2 H , with kxk D 1; consider the function ' x W Œ0; 1 ! R given
by ' x .t/ WD hf ..1 t/A C tB/x; xi: Since f is operator convex on I it follows
that ' x is convex on Œ0; 1 : Now, if we consider, for given t; s 2 .0; 1/;
t1 WD ts < s DW s1 and t2 WD t < t C .1 t/s DW s2 ;
then we have
' x .t1 / D hf ..1 ts/ A C tsB/ x; xi
and
' x .t2 / D hf ..1 t/A C tB/x; xi
giving that
' x .t1 / ' x .t2 /
D
t1 t2
f ..1 ts/ A C tsB/ f ..1 t/A C tB/
x; x :
t .s 1/
Also
' x .s1 / D hf ..1 s/ A C sB/ x; xi
and
' x .s2 / D hf ..1 t/ .1 s/ A C Œ.1 t/s C t B/ x; xi
giving that
' x .s1 / ' x .s2 /
s1 s2
f ..1 s/ A C sB/ f ..1 t/ .1 s/ A C Œ.1 t/s C t B/
x; x :
D
t .s 1/
Utilizing inequality (2.155) and multiplying with t .s 1/ < 0, we deduce the
following inequality in the operator order:
f ..1 ts/ A C tsB/ f ..1 t/A C tB/
f ..1 s/ A C sB/ f ..1 t/ .1 s/ A C Œ.1 t/s C t B/ :
(2.156)
Finally, by (2.154) and (2.156) we get the desired result (2.152).
Applying repeatedly the superadditivity property we have for ŒC; D ŒA; B
that
f .A; C I t/ C f .C; DI t/ C f .D; BI t/ f .A; BI t/
giving that
0 f .A; C I t/ C f .D; BI t/ f .A; BI t/ f .C; DI t/
which proves (2.153).
t
u
66
2 Inequalities of the Jensen Type
For t D
1
2
we consider the functional
f .A/ C f .B/
ACB
1
D
f
;
f .A; B/ WD f A; BI
2
2
2
which obviously inherits the superadditivity and monotonicity properties of the
functional f .; I t/ : We are able then to state the following:
Corollary 2.54 (Dragomir, 2010, [12]). Let f W I R ! R be an operator
convex function on the interval I . Then for each A; B two distinct selfadjoint
operators with the spectra in I we have the following bounds in the operator order:
C CB
ACB
ACC
Cf
f .C / D f
f
C 2ŒA;B
2
2
2
inf
(2.157)
and
C CD
f .C / C f .D/
f
2
2
C;D2ŒA;B
ACB
f .A/ C f .B/
:
f
D
2
2
sup
(2.158)
Proof. By the superadditivity of the functional f .; / we have for each C 2
ŒA; B that
ACB
2
f .A/ C f .C /
ACC
f .C / C f .B/
C CB
f
C
f
2
2
2
2
f .A/ C f .B/
f
2
which is equivalent with
f
ACC
2
Cf
C CB
2
f .C / f
ACB
:
2
(2.159)
Since the equality case in (2.159) is realized for either C D A or C D B we get the
desired bound (2.157).
The bound (2.158) is obvious by the monotonicity of the functional f .; / as a
function of interval.
t
u
Consider now the following functional:
f .A; BI t/ WD f .A/ C f .B/ f ..1 t/A C tB/ f ..1 t/B C tA/;
2.8 Hermite–Hadamard’s Type Inequalities for Operator Convex Functions
67
where, as above, f W I R ! R is a convex function on the convex set I and A; B
two distinct selfadjoint operators with the spectra in I while t 2 Œ0; 1 :
We notice that
f .A; BI t/ D f .B; AI t/ D f .A; BI 1 t/
and
f .A; BI t/ D f .A; BI t/ C f .A; BI 1 t/ 0
for any A; B and t 2 Œ0; 1 :
Therefore, we can state the following result as well:
Corollary 2.55 (Dragomir, 2010, [12]). Let f W I R ! R be an operator
convex function on the interval I . Then for each A; B two distinct selfadjoint
operators with the spectra in I; the functional f .; I t/ is operator superadditive
and operator non-decreasing as a function of interval.
In particular, if C 2 ŒA; B then we have the inequality
1
Œf ..1 t/A C tB/ C f ..1 t/B C tA/
2
1
Œf ..1 t/A C tC / C f ..1 t/C C tA/
2
1
C Œf ..1 t/C C tB/ C f ..1 t/B C tC / f .C /:
2
(2.160)
Also, if C; D 2 ŒA; B then we have the inequality
f .A/ C f .B/ f ..1 t/A C tB/ f ..1 t/B C tA/
f .C / C f .D/ f ..1 t/C C tD/ f ..1 t/C C tD/
(2.161)
for any t 2 Œ0; 1 :
Perhaps the most interesting functional we can consider is the following one:
f .A/ C f .B/
Θf .A; B/ D
2
Z
1
f ..1 t/A C tB/dt:
(2.162)
0
Notice that, by the second Hermite–Hadamard inequality for operator convex
functions we have that Θf .A; B/ 0 in the operator order.
We also observe that
Z
Θf .A; B/ D
0
1
Z
f .A; BI t/ dt D
1
0
f .A; BI 1 t/ dt:
(2.163)
68
2 Inequalities of the Jensen Type
Utilizing this representation, we can state the following result as well:
Corollary 2.56 (Dragomir, 2010, [12]). Let f W I R ! R be an operator
convex function on the interval I . Then for each A; B two distinct selfadjoint
operators with the spectra in I; the functional Θf .; / is operator superadditive and
operator non-decreasing as a function of interval. Moreover, we have the bounds in
the operator order
Z
1
Œf ..1 t/A C tC / C f ..1 t/C C tB/ P dt f .C /
inf
C 2ŒA;B
Z
0
1
f ..1 t/A C tB/dt
D
(2.164)
0
and
Z 1
f .C / C f .D/
f ..1 t/C C tD/dt
2
C;D2ŒA;B
0
Z 1
f .A/ C f .B/
f ..1 t/A C tB/dt:
D
2
0
sup
(2.165)
Remark 2.57. The above inequalities can be applied to various concrete operator
convex function of interest.
If we choose for instance inequality (2.165), then we get the following bounds in
the operator order:
Z 1
C r C Dr
..1 t/C C tD/r dt
2
C;D2ŒA;B
0
Z 1
r
r
A CB
..1 t/A C tB/r dt;
D
2
0
sup
(2.166)
where r 2 Œ1; 0 [ Œ1; 2 and A; B are selfadjoint operators with spectra in .0; 1/ :
If r 2 .0; 1/ then
Z
1
sup
C;D2ŒA;B
Z
1
D
0
..1 t/C C tD/r dt 0
..1 t/A C tB/r dt C r C Dr
2
Ar C B r
2
and A; B are selfadjoint operators with spectra in .0; 1/ :
(2.167)
References
69
We also have the operator bound for the logarithm
Z
1
ln..1 t/C C tD/dt sup
C;D2ŒA;B
Z
0
1
D
ln..1 t/A C tB/dt 0
ln.C / C ln.D/
2
ln.A/ C ln.B/
;
2
(2.168)
where A; B are selfadjoint operators with spectra in .0; 1/ :
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