Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex mappings
We obtain some inequalities of the Hermite – Hadamard type for $K$-bounded norm convex mappings between two normed spaces. The applications for twice differentiable functions in Banach spaces and functions defined by power series in Banach algebras are presented. Some discrete Jensen-type inequaliti...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507816469463040 |
|---|---|
| author | Dragomir, S. S. Драгомир, С. С. Драгомир, С. С. |
| author_facet | Dragomir, S. S. Драгомир, С. С. Драгомир, С. С. |
| author_sort | Dragomir, S. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:31:57Z |
| description | We obtain some inequalities of the Hermite – Hadamard type for $K$-bounded norm convex mappings between two normed
spaces. The applications for twice differentiable functions in Banach spaces and functions defined by power series in
Banach algebras are presented. Some discrete Jensen-type inequalities are also obtained. |
| first_indexed | 2026-03-24T02:15:19Z |
| format | Article |
| fulltext |
UDC 517.5
S. S. Dragomir (Coll. Eng. and Sci., Victoria Univ., Australia and School Comput. Sci.
and Appl. Math., Univ. Witwatersrand, Johannesburg, South Africa)
INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD TYPE
FOR \bfitK -BOUNDED NORM CONVEX MAPPINGS
IНТЕГРАЛЬНI НЕРIВНОСТI ТИПУ ЕРМIТА – АДАМАРА
ДЛЯ \bfitK -ОБМЕЖЕНИХ ВIДОБРАЖЕНЬ З ОПУКЛОЮ НОРМОЮ
We obtain some inequalities of the Hermite – Hadamard type for K -bounded norm convex mappings between two normed
spaces. The applications for twice differentiable functions in Banach spaces and functions defined by power series in
Banach algebras are presented. Some discrete Jensen-type inequalities are also obtained.
Отримано деякi нерiвностi типу Ермiта – Адамара для K -обмежених вiдображень з опуклою нормою мiж двома
нормованими просторами. Наведено застосування до двiчi диференцiйовних функцiй у банахових просторах та
функцiй, що визначенi степеневими рядами в банахових алгебрах. Отримано також деякi нерiвностi типу Джексона.
1. Introduction. Let \scrB (H) be the Banach algebra of bounded linear operators on a complex Hilbert
space H. The absolute value of an operator A is the positive operator | A| defined as | A| := (A\ast A)1/2 .
One of the central problems in perturbation theory is to find bounds for
\| f(A) - f(B)\|
in terms of \| A - B\| for different classes of measurable functions f for which the function of operator
can be defined. For some results on this topic, see [5, 34] and the references therein.
It is known that [4] in the infinite-dimensional case the map f(A) := | A| is not Lipschitz
continuous on \scrB (H) with the usual operator norm, i.e., there is no constant L > 0 such that
\| | A| - | B| \| \leq L\| A - B\|
for any A,B \in \scrB (H).
However, as shown by Farforovskaya in [32, 33] and Kato in [39], the following inequality holds:
\| | A| - | B| \| \leq 2
\pi
\| A - B\|
\biggl(
2 + \mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
\| A\| + \| B\|
\| A - B\|
\biggr) \biggr)
for any A,B \in \scrB (H) with A \not = B.
If the operator norm is replaced with Hilbert – Schmidt norm \| C\| HS := (\mathrm{t}\mathrm{r}C\ast C)1/2 of an
operator C, then the following inequality is true [2]:
\| | A| - | B| \| HS \leq
\surd
2\| A - B\| HS
for any A,B \in \scrB (H).
The coefficient
\surd
2 is best possible for a general A and B. If A and B are restricted to be
self-adjoint, then the best coefficient is 1.
It has been shown in [4] that, if A is an invertible operator, then for all operators B in a
neighborhood of A we have
c\bigcirc S. S. DRAGOMIR, 2016
1330 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1331
\| | A| - | B| \| \leq a1\| A - B\| + a2\| A - B\| 2 +O
\bigl(
\| A - B\| 3
\bigr)
,
where
a1 =
\bigm\| \bigm\| A - 1
\bigm\| \bigm\| \| A\| and a2 =
\bigm\| \bigm\| A - 1
\bigm\| \bigm\| +
\bigm\| \bigm\| A - 1
\bigm\| \bigm\| 3 \| A\| 2.
In [3] the author also obtained the Lipschitz-type inequality
\| f(A) - f(B)\| \leq f \prime (a)\| A - B\| ,
where f is an operator monotone function on (0,\infty ) and A,B \geq aIH > 0.
Let (X; \| \cdot \| X) and (Y ; \| \cdot \| Y ) be two Banach spaces over the complex number field \BbbC . Let C
be a convex set in X. For any mapping F : C \subset X \rightarrow Y we can consider the associated functions
\Phi F,x,y,\lambda , \Psi F,x,y,\lambda : [0, 1] \rightarrow Y, where x, y \in C, \lambda \in [0, 1], defined by [25]
\Phi F,x,y,\lambda (t) := (1 - \lambda )F [(1 - t)((1 - \lambda )x+ \lambda y) + ty] + \lambda F [(1 - t)x+ t((1 - \lambda )x+ \lambda y)]
and
\Psi F,x,y,\lambda (t) := (1 - \lambda )F [(1 - t)((1 - \lambda )x+ \lambda y) + ty] + \lambda F [tx+ (1 - t)((1 - \lambda )x+ \lambda y)].
We say that the mapping F : B \subset X \rightarrow Y is Lipschitzian with the constant L > 0 on the subset
B of X if
\| F (x) - F (y)\| Y \leq L\| x - y\| X for any x, y \in B.
The following result holds [25]:
Theorem 1.1. Let F : C \subset X \rightarrow Y be a Lipschitzian mapping with the constant L > 0 on the
convex subset C of X. If x, y \in C, then we have\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \Lambda F,x,y,\lambda (t) -
1\int
0
F [sy + (1 - s)x] ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq
\leq 2L
\Biggl[
1
4
+
\biggl(
t - 1
2
\biggr) 2
\Biggr] \Biggl[
1
4
+
\biggl(
\lambda - 1
2
\biggr) 2
\Biggr]
\| x - y\| X (1.1)
for any t \in [0, 1] and \lambda \in [0, 1], where \Lambda F,x,y,\lambda = \Phi F,x,y,\lambda or \Lambda F,x,y,\lambda = \Psi F,x,y,\lambda .
If we take in (1.1) \Lambda F,x,y,\lambda = \Phi F,x,y,\lambda , \lambda =
1
2
, then we get\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 12
\biggl(
F
\biggl[
(1 - t)
x+ y
2
+ ty
\biggr]
+ F
\biggl[
(1 - t)x+ t
x+ y
2
\biggr] \biggr)
-
1\int
0
F [sy + (1 - s)x]ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq 1
2
L
\Biggl[
1
4
+
\biggl(
t - 1
2
\biggr) 2
\Biggr]
\| x - y\| X
for any x, y \in C and t \in [0, 1].
If we take in (1.1) \Lambda F,x,y,\lambda = \Psi F,x,y,\lambda , \lambda =
1
2
, then we obtain
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1332 S. S. DRAGOMIR\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 12
\biggl(
F
\biggl[
(1 - t)
x+ y
2
+ ty
\biggr]
+ F
\biggl[
tx+ (1 - t)
x+ y
2
\biggr] \biggr)
-
1\int
0
F [sy + (1 - s)x]ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq
\leq 1
2
L
\Biggl[
1
4
+
\biggl(
t - 1
2
\biggr) 2
\Biggr]
\| x - y\| X
for any t \in [0, 1] and x, y \in C.
We also have the simpler inequalities\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 12
\biggl[
F
\biggl(
3x+ y
4
\biggr)
+ F
\biggl(
x+ 3y
4
\biggr) \biggr]
-
1\int
0
F [sy + (1 - s)x] ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
8
L\| x - y\| X , (1.2)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F
\biggl(
x+ y
2
\biggr)
-
1\int
0
F [sy + (1 - s)]ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
4
L\| x - y\| X (1.3)
and \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 12[F (x) + F (y)] -
1\int
0
F [sy + (1 - s)x]ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
4
L\| x - y\| X (1.4)
for any x, y \in C. The constants
1
8
and
1
4
are best possible.
The inequalities (1.3) and (1.4) are the corresponding versions of Hermite – Hadamard inequalities
for Lipschitzian functions. The scalar cases were obtained in [12] and [43]. For Hermite – Hadamard’s
type inequalities, see for instance [10, 12, 13, 35, 37, 38, 40, 42, 43, 46 – 50] and the references
therein.
From (1.1) we also have the Ostrowski’s inequality\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F [ty + (1 - t)x] -
1\int
0
F [sy + (1 - s)x]ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq L
\Biggl[
1
4
+
\biggl(
t - 1
2
\biggr) 2
\Biggr]
\| x - y\| X
for any t \in [0, 1] and x, y \in C. For Ostrowski’s type inequalities for the Lebesgue integral, see [1,
8, 9, 15 – 30]. Inequalities for the Riemann – Stieltjes integral may be found in [17, 19] while the
generalization for isotonic functionals was provided in [20]. For the case of functions of self-adjoint
operators on complex Hilbert spaces, see the recent monograph [23].
Motivated by the above results, we introduce here a class of functions that extends the concept
of Lipschitzian function to power two of norm difference and called them K -bounded norm convex
functions. Comprehensive examples of such functions are given. Integral inequalities of Hermite –
Hadamard type are obtained and applications for discrete inequalities of Jensen type are provided as
well.
2. \bfitK -bounded norm convex mappings. Let (X; \| \cdot \| X) and (Y ; \| \cdot \| Y ) be two normed linear
spaces over the complex number field \BbbC . Let C be a convex set in X. We consider the following
class of functions:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1333
Definition 2.1. A mapping F : C \subset X \rightarrow Y is called K -bounded norm convex, for some given
K > 0 if it satisfies the condition
\| (1 - \lambda )F (x) + \lambda F (y) - F ((1 - \lambda )x+ \lambda y)\| Y \leq 1
2
K\lambda (1 - \lambda )\| x - y\| 2X (2.1)
for any x, y \in C and \lambda \in [0, 1]. For simplicity, we denote this by F \in \scrB \scrN K(C).
We have from (2.1) for \lambda =
1
2
the Jensen’s inequality\bigm\| \bigm\| \bigm\| \bigm\| F (x) + F (y)
2
- F
\biggl(
x+ y
2
\biggr) \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
8
K\| x - y\| 2X (2.2)
for any x, y \in C.
We observe that \scrB \scrN K(C) is a convex subset in the linear space of all functions defined on C
and with values in Y.
We observe also that, by the triangle inequality, we have
\| F ((1 - \lambda )x+ \lambda y)\| Y - \| (1 - \lambda )F (x) + \lambda F (y)\| Y \leq
\leq \| (1 - \lambda )F (x) + \lambda F (y) - F ((1 - \lambda )x+ \lambda y)\| Y
which, by the triangle inequality, gives
\| F ((1 - \lambda )x+ \lambda y)\| Y \leq 1
2
K\lambda (1 - \lambda )\| x - y\| 2X + (1 - \lambda )\| F (x)\| Y + \lambda \| F (y)\| Y (2.3)
for any x, y \in C and \lambda \in [0, 1].
Now, if the function t \mapsto \rightarrow \| F ((1 - \lambda )x + \lambda y)\| Y , for some x, y \in C, is Lebesgue integrable on
[0, 1], then by taking the integral in (2.3) we get
1\int
0
\| F ((1 - \lambda )x+ \lambda y)\| Y d\lambda \leq 1
12
K\| x - y\| 2X +
1
2
[\| F (x)\| Y + \| F (y)\| Y ] . (2.4)
If we assume continuity for the function F on C in the norm topology of (X; \| \cdot \| X), then the
inequality (2.4) holds for any x, y \in C. Moreover, if we assume that (Y ; \| \cdot \| Y ) is a Banach space
and F is continuos on C, then we have the generalized triangle inequality\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
F ((1 - \lambda )x+ \lambda y)d\lambda
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq
1\int
0
\| F ((1 - \lambda )x+ \lambda y)\| Y d\lambda ,
and by (2.4) we obtain\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
F ((1 - \lambda )x+ \lambda y)d\lambda
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
12
K\| x - y\| 2X +
1
2
[\| F (x)\| Y + \| F (y)\| Y ]
for any x, y \in C.
We can improve this result as follows.
Theorem 2.1. (X; \| \cdot \| X) and (Y ; \| \cdot \| Y ) be two normed linear spaces over the complex number
field \BbbC with Y complete. Assume that the mapping F : C \subset X \rightarrow Y is continuous on the convex
set C in the norm topology. If F \in \scrB \scrN K(C) for some K > 0, then we have
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1334 S. S. DRAGOMIR\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| F (x) + F (y)
2
-
1\int
0
F ((1 - \lambda )x+ \lambda y)d\lambda
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
12
K\| x - y\| 2X (2.5)
and \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
F ((1 - \lambda )x+ \lambda y)d\lambda - F
\biggl(
x+ y
2
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
24
K\| x - y\| 2X (2.6)
for any x, y \in C. The constants
1
12
and
1
24
are best possible.
Proof. From (2.1) we get successively\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
[(1 - \lambda )F (x) + \lambda F (y) - F ((1 - \lambda )x+ \lambda y)]d\lambda
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq
\leq
1\int
0
\| (1 - \lambda )F (x) + \lambda F (y) - F ((1 - \lambda )x+ \lambda y)\| Y d\lambda \leq
\leq 1
2
K\| x - y\| 2X
1\int
0
\lambda (1 - \lambda )d\lambda ,
which produces the desired result (2.5).
Utilising (2.2) we have\bigm\| \bigm\| \bigm\| \bigm\| F ((1 - \lambda )x+ \lambda y) + F (\lambda x+ (1 - \lambda )y)
2
- F
\biggl(
x+ y
2
\biggr) \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq
\leq 1
8
K\| (1 - \lambda )x+ \lambda y - \lambda x - (1 - \lambda )y\| 2X =
=
1
8
K(1 - 2\lambda )2\| x - y\| 2X =
1
2
K
\biggl(
\lambda - 1
2
\biggr) 2
\| x - y\| 2X (2.7)
for any x, y \in C and \lambda \in [0, 1].
Integrating in (2.7) we obtain\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
\biggl[
F ((1 - \lambda )x+ \lambda y) + F (\lambda x+ (1 - \lambda )y)
2
- F
\biggl(
x+ y
2
\biggr) \biggr]
d\lambda
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq
\leq
1\int
0
\bigm\| \bigm\| \bigm\| \bigm\| F ((1 - \lambda )x+ \lambda y) + F (\lambda x+ (1 - \lambda )y)
2
- F
\biggl(
x+ y
2
\biggr) \bigm\| \bigm\| \bigm\| \bigm\|
Y
d\lambda \leq
\leq 1
2
K\| x - y\| 2X
1\int
0
\biggl(
\lambda - 1
2
\biggr) 2
d\lambda =
1
24
K\| x - y| 2X (2.8)
and since
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1335
1\int
0
F ((1 - \lambda )x+ \lambda y)d\lambda =
1\int
0
F (\lambda x+ (1 - \lambda )y)d\lambda ,
then from (2.8) we get (2.6).
Now, consider the function F0 : H \rightarrow \BbbR , F0(x) = \| x\| 2, where (H, \langle ., .\rangle ) is a complex inner
product space. If x, y \in H and \lambda \in [0, 1], then
(1 - \lambda )F0(x) + \lambda F0(y) - F0((1 - \lambda )x+ \lambda y) =
= (1 - \lambda )\| x\| 2 + \lambda \| y\| 2 - \| (1 - \lambda )x+ \lambda y\| 2 =
= (1 - \lambda )\| x\| 2 + \lambda \| y\| 2 - (1 - \lambda )2\| x\| 2 - 2(1 - \lambda )\lambda \mathrm{R}\mathrm{e} \langle x, y\rangle - \lambda 2\| y\| 2 =
= (1 - \lambda )\lambda
\bigl[
\| x\| 2 - 2\mathrm{R}\mathrm{e} \langle x, y\rangle + \| y\| 2
\bigr]
= (1 - \lambda )\lambda \| x - y\| 2
showing that F0 is continuous and K -bounded norm convex with K = 2 on H.
We have
1\int
0
F0((1 - \lambda )x+ \lambda y)d\lambda =
1\int
0
\| (1 - \lambda )x+ \lambda y\| 2d\lambda =
=
1\int
0
\bigl[
(1 - \lambda )2\| x\| 2 + 2(1 - \lambda )\lambda \mathrm{R}\mathrm{e} \langle x, y\rangle + \lambda 2\| y\| 2
\bigr]
d\lambda =
=
1
3
\bigl[
\| x\| 2 +\mathrm{R}\mathrm{e} \langle x, y\rangle + \| y\| 2
\bigr]
for any x, y \in H.
Therefore
F0(x) + F0(y)
2
-
1\int
0
F0((1 - \lambda )x+ \lambda y)d\lambda =
=
1
2
\bigl[
\| x\| 2 + \| y\| 2
\bigr]
- 1
3
\bigl[
\| x\| 2 +\mathrm{R}\mathrm{e} \langle x, y\rangle + \| y\| 2
\bigr]
=
1
6
\| x - y\| 2
showing that we have the same quantity
1
6
\| x - y\| 2 in both sides of (2.5).
We also have
1\int
0
F0((1 - \lambda )x+ \lambda y)d\lambda - F0
\biggl(
x+ y
2
\biggr)
=
=
1
3
\bigl[
\| x\| 2 +\mathrm{R}\mathrm{e} \langle x, y\rangle + \| y\| 2
\bigr]
- 1
4
\bigl[
\| x\| 2 + 2\mathrm{R}\mathrm{e} \langle x, y\rangle + \| y\| 2
\bigr]
=
1
12
\| x - y\| 2
showing that we have the same quantity
1
12
\| x - y\| 2 in both sides of (2.6).
Theorem 2.1 is proved.
3. Some examples in Banach algebras. Let \scrB be an algebra. An algebra norm on \scrB is a map
\| \cdot \| : \scrB \rightarrow [0,\infty ) such that (\scrB , \| \cdot \| ) is a normed space, and, further,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1336 S. S. DRAGOMIR
\| ab\| \leq \| a\| \| b\|
for any a, b \in \scrB . The normed algebra (\scrB , \| \cdot \| ) is a Banach algebra if \| \cdot \| is a complete norm.
We assume that the Banach algebra is unital, this means that \scrB has an identity 1 and that \| 1\| = 1.
Let \scrB be a unital algebra. An element a \in \scrB is invertible if there exists an element b \in \scrB with
ab = ba = 1. The element b is unique; it is called the inverse of a and written a - 1 or
1
a
. The set of
invertible elements of \scrB is denoted by \mathrm{I}\mathrm{n}\mathrm{v}\scrB . If a, b \in \mathrm{I}\mathrm{n}\mathrm{v}\scrB then ab \in \mathrm{I}\mathrm{n}\mathrm{v}\scrB and (ab) - 1 = b - 1a - 1.
For a unital Banach algebra we also have
(i) if a \in \scrB and \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| an\| 1/n < 1, then 1 - a \in \mathrm{I}\mathrm{n}\mathrm{v}\scrB ;
(ii) \{ a \in \scrB : \| 1 - b\| < 1\} \subset \mathrm{I}\mathrm{n}\mathrm{v}\scrB ;
(iii) \mathrm{I}\mathrm{n}\mathrm{v}\scrB is an open subset of \scrB ;
(iv) the map \mathrm{I}\mathrm{n}\mathrm{v}\scrB \ni a \mapsto - \rightarrow a - 1 \in \mathrm{I}\mathrm{n}\mathrm{v}\scrB is continuous.
The resolvent set of a \in \scrB is defined by
\rho (a) := \{ z \in \BbbC : z1 - a \in \mathrm{I}\mathrm{n}\mathrm{v}\scrB \} ;
the spectrum of a is \sigma (a), the complement of \rho (a) in \BbbC , and the resolvent function of a is Ra :
\rho (a) \rightarrow \mathrm{I}\mathrm{n}\mathrm{v}\scrB ,
Ra(z) := (z1 - a) - 1.
For each z, w \in \rho (a) we get the identity
Ra (w) - Ra (z) = (z - w)Ra (z)Ra (w) .
We also obtain
\sigma (a) \subset \{ z \in \BbbC : \| z\| \leq \| a\| \} .
The spectral radius of a is defined as
\nu (a) = \mathrm{s}\mathrm{u}\mathrm{p}\{ \| z\| : z \in \sigma (a)\} .
If a, b are commuting elements in \scrB , i.e., ab = ba, then
\nu (ab) \leq \nu (a)\nu (b) and \nu (a+ b) \leq \nu (a) + \nu (b).
Let \scrB be a unital Banach algebra and a \in \scrB . Then
(i) the resolvent set \rho (a) is open in \BbbC ;
(ii) for any bounded linear functionals \lambda : \scrB \rightarrow \BbbC , the function \lambda \circ Ra is analytic on \rho (a);
(iii) the spectrum \sigma (a) is compact and nonempty in \BbbC ;
(iv) we have
\nu (a) = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| an\| 1/n .
Let f be an analytic functions on the open disk D(0, R) given by the power series
f(z) :=
\infty \sum
j=0
\alpha jz
j (\| z\| < R).
If \nu (a)<R, then the series
\sum \infty
j=0
\alpha ja
j converges in the Banach algebra \scrB because
\sum \infty
j=0
| \alpha j | \| aj\| <
< \infty , and we can define f(a) to be its sum. Clearly f(a) is well defined and there are many examples
of important functions on a Banach algebra \scrB that can be constructed in this way. For instance, the
exponential map on \scrB denoted \mathrm{e}\mathrm{x}\mathrm{p} and defined as
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INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1337
\mathrm{e}\mathrm{x}\mathrm{p} a :=
\infty \sum
j=0
1
j!
aj for each a \in \scrB .
If \scrB is not commutative, then many of the familiar properties of the exponential function from the
scalar case do not hold. The following key formula is valid, however with the additional hypothesis
of commutativity for a and b from \scrB
\mathrm{e}\mathrm{x}\mathrm{p}(a+ b) = \mathrm{e}\mathrm{x}\mathrm{p}(a) \mathrm{e}\mathrm{x}\mathrm{p}(b).
In a general Banach algebra \scrB it is difficult to determine the elements in the range of the exponential
map \mathrm{e}\mathrm{x}\mathrm{p}(\scrB ), i.e., the element which have a "logarithm". However, it is easy to see that if a is an
element in B such that \| 1 - a\| < 1, then a is in \mathrm{e}\mathrm{x}\mathrm{p}(\scrB ). That follows from the fact that if we set
b = -
\infty \sum
n=1
1
n
(1 - a)n,
then the series converges absolutely and, as in the scalar case, substituting this series into the series
expansion for \mathrm{e}\mathrm{x}\mathrm{p}(b) yields \mathrm{e}\mathrm{x}\mathrm{p}(b) = a.
Concerning other basic definitions and facts in the theory of Banach algebras, the reader can
consult the classical books [31] and [45].
Now, by the help of power series f(\lambda ) =
\sum \infty
n=0
\alpha n\lambda
n we can naturally construct another power
series which will have as coefficients the absolute values of the coefficients of the original series,
namely, fabs(\lambda ) :=
\sum \infty
n=0
\| \alpha n\| \lambda n. It is obvious that this new power series will have the same radius
of convergence as the original series. We also notice that if all coefficients \alpha n \geq 0, then fabs = f.
The following result provides a class of functions that are K -bounded norm convex on closed
balls from Banach algebras.
Theorem 3.1. Let f(z) =
\sum \infty
n=0
\alpha nz
n be a function defined by power series with complex
coefficients and convergent on the open disk D(0, R) \subset \BbbC , R > 0. For any x, y \in \scrB with
\| x\| , \| y\| \leq M < R, M > 0 we have that
\| (1 - \lambda )f(x) + \lambda f(y) - f((1 - \lambda )x+ \lambda y)\| \leq 1
2
\lambda (1 - \lambda ) f \prime \prime
abs(M)\| x - y\| 2 (3.1)
for any \lambda \in [0, 1].
In other words the function f : B(0,M) \subset \scrB \rightarrow \scrB , where B(0,M) is the closed ball \{ x \in
\in \scrB , \| x\| \leq M\} defined by f(x) =
\sum \infty
n=0
\alpha nx
n, x \in B(0,M) is K -bounded norm convex with
K = f \prime \prime
abs(M).
Proof. We use the identity (see, for instance, [6, p. 254])
an - bn =
n - 1\sum
j=0
an - 1 - j(a - b)bj (3.2)
that holds for any a, b \in \scrB and n \geq 1.
Let x, y \in \scrB . By (3.2) we have
[(1 - \lambda )x+ \lambda y]n - xn = \lambda
n - 1\sum
j=0
[(1 - \lambda )x+ \lambda y]n - 1 - j(y - x)xj (3.3)
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1338 S. S. DRAGOMIR
and
[(1 - \lambda )x+ \lambda y]n - yn = - (1 - \lambda )
n - 1\sum
j=0
[(1 - \lambda )x+ \lambda y]n - 1 - j(y - x)yj (3.4)
for n \geq 1 and \lambda \in [0, 1].
Multiply (3.3) by 1 - \lambda and (3.4) by \lambda and add the obtained equalities to get
[(1 - \lambda )x+ \lambda y]n - (1 - \lambda )xn - \lambda yn =
= \lambda (1 - \lambda )
n - 1\sum
j=0
[(1 - \lambda )x+ \lambda y]n - 1 - j(y - x)
\bigl(
xj - yj
\bigr)
=
= \lambda (1 - \lambda )
n - 1\sum
j=1
[(1 - \lambda )x+ \lambda y]n - 1 - j(y - x)
\bigl(
xj - yj
\bigr)
(3.5)
for n \geq 2 and \lambda \in [0, 1].
If j \geq 1 we also obtain
xj - yj =
j - 1\sum
\ell =0
xj - 1 - \ell (x - y) y\ell
and by (3.5) we have
(1 - \lambda )xn + \lambda yn - [(1 - \lambda )x+ \lambda y]n =
= \lambda (1 - \lambda )
n - 1\sum
j=1
j - 1\sum
\ell =0
[(1 - \lambda )x+ \lambda y]n - 1 - j(y - x)xj - 1 - \ell (x - y)y\ell (3.6)
for n \geq 2 and \lambda \in [0, 1], which is an equality of interest in itself.
Let m \geq 2 and x, y \in \scrB . Then, by utilizing (3.6), we get
(1 - \lambda )
m\sum
n=0
anx
n + \lambda
m\sum
n=0
any
n -
m\sum
n=0
an[(1 - \lambda )x+ \lambda y]n =
=
m\sum
n=0
an[(1 - \lambda )xn + \lambda yn - [(1 - \lambda )x+ \lambda y]n] =
=
m\sum
n=2
an[(1 - \lambda )xn + \lambda yn - [(1 - \lambda )x+ \lambda y]n] =
= \lambda (1 - \lambda )
m\sum
n=2
an
\left( n - 1\sum
j=1
j - 1\sum
\ell =0
[(1 - \lambda )x+ \lambda y]n - 1 - j(y - x)xj - 1 - \ell (x - y)y\ell
\right) (3.7)
for all m \geq 2, x, y \in \scrB and \lambda \in [0, 1].
Taking the norm in (3.7) and using repeatedly the generalized triangle inequality we have
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INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1339\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| (1 - \lambda )
m\sum
n=0
anx
n + \lambda
m\sum
n=0
any
n -
m\sum
n=0
an[(1 - \lambda )x+ \lambda y]n
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq \lambda (1 - \lambda )
m\sum
n=2
| an|
\left( n - 1\sum
j=1
j - 1\sum
\ell =0
\bigm\| \bigm\| \bigm\| [(1 - \lambda )x+ \lambda y]n - 1 - j(y - x)xj - 1 - \ell (x - y)y\ell
\bigm\| \bigm\| \bigm\|
\right) . (3.8)
If \| x\| , \| y\| \leq M < R, then \| (1 - \lambda )x + \lambda y\| \leq M for \lambda \in [0, 1] and using the Banach algebra
properties we obtain \bigm\| \bigm\| \bigm\| [(1 - \lambda )x+ \lambda y]n - 1 - j(y - x)xj - 1 - \ell (x - y)y\ell
\bigm\| \bigm\| \bigm\| \leq
\leq \| (1 - \lambda )x+ \lambda y\| n - 1 - j\| y - x\| \| x\| j - 1 - \ell \| x - y\| \| y\| \ell =
= \| y - x\| 2\| (1 - \lambda )x+ \lambda y\| n - 1 - j\| x\| j - 1 - \ell \| y\| \ell \leq
\leq \| y - x\| 2Mn - 1 - jM j - 1 - \ell M \ell = \| y - x\| 2Mn - 2 (3.9)
for n \geq 2.
Therefore, by (3.8) and (3.9) we get\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| (1 - \lambda )
m\sum
n=0
anx
n + \lambda
m\sum
n=0
any
n -
m\sum
n=0
an[(1 - \lambda )x+ \lambda y]n
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq \lambda (1 - \lambda )
m\sum
n=2
| an|
\left( n - 1\sum
j=1
j - 1\sum
\ell =0
\| y - x\| 2Mn - 2
\right) =
= \lambda (1 - \lambda )\| y - x\| 2
m\sum
n=2
\| an\| Mn - 2
n - 1\sum
j=1
j =
=
1
2
\lambda (1 - \lambda )\| y - x\| 2
m\sum
n=2
n(n - 1)\| an\| Mn - 2 (3.10)
for any \| x\| , \| y\| \leq M < R, m \geq 2 and \lambda \in [0, 1].
Since the series whose partial sums involved in (3.10) are convergent and
\infty \sum
n=0
anx
n = f(x),
\infty \sum
n=0
any
n = f(y),
m\sum
n=0
an[(1 - \lambda )x+ \lambda y]n = f((1 - \lambda )x+ \lambda y),
\infty \sum
n=2
n(n - 1)\| an\| Mn - 2 = f \prime \prime
abs(M),
then by letting m \rightarrow \infty in (3.10) we deduce the desired result (3.1).
Theorem 3.1 is proved.
Corollary 3.1. With the assumptions from Theorem 3.1 we have the inequalities
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1340 S. S. DRAGOMIR\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f(x) + f(y)
2
-
1\int
0
f((1 - \lambda )x+ \lambda y)d\lambda
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq 1
12
f \prime \prime
abs(M)\| x - y\| 2 (3.11)
and \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
f((1 - \lambda )x+ \lambda y)d\lambda - f
\biggl(
x+ y
2
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq 1
24
f \prime \prime
abs(M)\| x - y\| 2 (3.12)
for any x, y \in \scrB with \| x\| , \| y\| \leq M < R, M > 0.
The constants
1
12
and
1
24
are best possible.
It is known that if x and y are commuting, i.e., xy = yx, then the exponential function satisfies
the property
\mathrm{e}\mathrm{x}\mathrm{p}(x) \mathrm{e}\mathrm{x}\mathrm{p}(y) = \mathrm{e}\mathrm{x}\mathrm{p}(y) \mathrm{e}\mathrm{x}\mathrm{p}(x) = \mathrm{e}\mathrm{x}\mathrm{p}(x+ y).
Also, if z is invertible and a, b \in \BbbR with a < b, then
b\int
a
\mathrm{e}\mathrm{x}\mathrm{p}(tz)dt = z - 1[\mathrm{e}\mathrm{x}\mathrm{p}(bz) - \mathrm{e}\mathrm{x}\mathrm{p}(az)].
Therefore, if x and y are commuting and y - x is invertible, then
1\int
0
\mathrm{e}\mathrm{x}\mathrm{p}((1 - s)x+ sy)ds =
1\int
0
\mathrm{e}\mathrm{x}\mathrm{p}(s(y - x)) \mathrm{e}\mathrm{x}\mathrm{p}(x)ds =
=
\left( 1\int
0
\mathrm{e}\mathrm{x}\mathrm{p}(s(y - x))ds
\right) \mathrm{e}\mathrm{x}\mathrm{p}(x) =
= (y - x) - 1[\mathrm{e}\mathrm{x}\mathrm{p}(y - x) - 1] \mathrm{e}\mathrm{x}\mathrm{p}(x) = (y - x) - 1[\mathrm{e}\mathrm{x}\mathrm{p}(y) - \mathrm{e}\mathrm{x}\mathrm{p}(x)],
and by (3.11) and (3.12) we get\bigm\| \bigm\| \bigm\| \bigm\| \mathrm{e}\mathrm{x}\mathrm{p}(x) + \mathrm{e}\mathrm{x}\mathrm{p}(y)
2
- (y - x) - 1[\mathrm{e}\mathrm{x}\mathrm{p}(y) - \mathrm{e}\mathrm{x}\mathrm{p}(x)]
\bigm\| \bigm\| \bigm\| \bigm\| \leq 1
12
\mathrm{e}\mathrm{x}\mathrm{p}(M)\| x - y\| 2
and \bigm\| \bigm\| \bigm\| \bigm\| (y - x) - 1[\mathrm{e}\mathrm{x}\mathrm{p}(y) - \mathrm{e}\mathrm{x}\mathrm{p}(x)] - \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
x+ y
2
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| \leq 1
24
\mathrm{e}\mathrm{x}\mathrm{p}(M)\| x - y\| 2
provided \| x\| , \| y\| \leq M, M > 0.
4. Case of twice differentiable mappings in normed spaces. We first recall some results
concerning Taylor’s formula for differentiable mappings between two normed spaces, see for instance
[11] for the basic definitions and results.
Lemma 4.1 (Taylor’s formula, Lagrange’s remainder [11, p. 110, 111]) . Let (X, \| \cdot \| X) and
(Y, \| \cdot \| Y ) be two normed linear spaces, \Omega an open subset of X and f : \Omega \rightarrow Y a(k + 1)-dif-
ferentiable mapping on \Omega with k \geq 0. Suppose that x, y \in \Omega are such that the segment [x, y] :=
:= \{ (1 - \lambda )x+ \lambda y, \lambda \in [0, 1]\} is contained in \Omega . Then
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INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1341
f(y) = f(x) + f (1)(x)(y - x) +
1
2!
f (2)(x)(y - x, y - x) + . . .
. . .+
1
k!
f (k)(x)(y - x, . . . , y - x) +Rk(x, y), (4.1)
where
\| Rk(x, y)\| Y \leq 1
(k + 1)!
\| y - x\| k+1
X \mathrm{s}\mathrm{u}\mathrm{p}
\lambda \in [0,1]
\bigm\| \bigm\| \bigm\| f (k+1)((1 - \lambda )x+ \lambda y)
\bigm\| \bigm\| \bigm\|
\scrL (Xk+1;Y )
.
We observe that if \Omega is open and convex, then the equality (4.1) holds for any x, y \in \Omega . In this
case we also have the bound
\| Rk(x, y)\| Y \leq 1
(k + 1)!
\| y - x\| k+1
X \mathrm{s}\mathrm{u}\mathrm{p}
z\in \Omega
\bigm\| \bigm\| \bigm\| f (k+1)(z)
\bigm\| \bigm\| \bigm\|
\scrL (Xk+1;Y )
for any x, y \in \Omega .
We can prove the following result:
Theorem 4.1. Let (X, \| \cdot \| X) and (Y, \| \cdot \| Y ) be two normed linear spaces, C an open convex
subset of X and f : C \rightarrow Y a twice-differentiable mapping on C. Then for any x, y \in C and
\lambda \in [0, 1] we have
\| (1 - \lambda )f(x) + \lambda f(y) - f((1 - \lambda )x+ \lambda y)\| Y \leq 1
2
K\lambda (1 - \lambda )\| y - x\| 2X , (4.2)
where
K := \mathrm{s}\mathrm{u}\mathrm{p}
z\in C
\bigm\| \bigm\| f \prime \prime (z)
\bigm\| \bigm\|
\scrL (X2;Y )
(4.3)
is assumed to be finite.
Proof. Using the above Lemma 4.1 we can state that\bigm\| \bigm\| f(u) - f(v) - f \prime (v)(u - v)
\bigm\| \bigm\|
F
\leq 1
2
K\| u - v\| 2X (4.4)
for any u, v \in C, where K is given by (4.3).
Let x, y \in C and \lambda \in [0, 1]. By (4.4) we have\bigm\| \bigm\| f(x) - f((1 - \lambda )x+ \lambda y) - \lambda f \prime ((1 - \lambda )x+ \lambda y)(x - y)
\bigm\| \bigm\|
Y
\leq 1
2
K\lambda 2\| y - x\| 2X (4.5)
and \bigm\| \bigm\| f(y) - f((1 - \lambda )x+ \lambda y) - (1 - \lambda )f \prime ((1 - \lambda )x+ \lambda y)(y - x)
\bigm\| \bigm\|
Y
\leq
\leq 1
2
K(1 - \lambda )2\| y - x\| 2X . (4.6)
Multiply (4.5) by 1 - \lambda and (4.6) by \lambda and add the obtained inequalities to get
(1 - \lambda )
\bigm\| \bigm\| f(x) - f((1 - \lambda )x+ \lambda y) - \lambda f \prime ((1 - \lambda )x+ \lambda y)(x - y)
\bigm\| \bigm\|
Y
+
+\lambda
\bigm\| \bigm\| f(y) - f((1 - \lambda )x+ \lambda y) + (1 - \lambda )f \prime ((1 - \lambda )x+ \lambda y)(x - y)
\bigm\| \bigm\|
Y
\leq
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1342 S. S. DRAGOMIR
\leq 1
2
K\lambda 2(1 - \lambda )\| y - x\| 2X +
1
2
K(1 - \lambda )2\lambda \| y - x\| 2X =
1
2
K\lambda (1 - \lambda )\| y - x\| 2X . (4.7)
By the triangle inequality we also obtain
\| (1 - \lambda )f(x) + \lambda f(y) - f((1 - \lambda )x+ \lambda y)\| Y \leq
\leq (1 - \lambda )
\bigm\| \bigm\| f(x) - f((1 - \lambda )x+ \lambda y) - \lambda f \prime ((1 - \lambda )x+ \lambda y)(x - y)
\bigm\| \bigm\|
Y
+
+\lambda
\bigm\| \bigm\| f(y) - f((1 - \lambda )x+ \lambda y) + (1 - \lambda )f \prime ((1 - \lambda )x+ \lambda y)(x - y)
\bigm\| \bigm\|
Y
(4.8)
for any x, y \in C and \lambda \in [0, 1].
Making use of (4.7) and (4.8) we deduce the desired result (4.2).
Theorem 4.1 is proved.
Corollary 4.1. With the assumptions from Theorem 4.1 we have the inequalities\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| f(x) + f(y)
2
-
1\int
0
f((1 - \lambda )x+ \lambda y)d\lambda
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
12
\mathrm{s}\mathrm{u}\mathrm{p}
z\in C
\bigm\| \bigm\| f \prime \prime (z)
\bigm\| \bigm\|
\scrL (X2;Y )
\| x - y\| 2X
and \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
f((1 - \lambda )x+ \lambda y)d\lambda - f
\biggl(
x+ y
2
\biggr) \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq 1
24
\mathrm{s}\mathrm{u}\mathrm{p}
z\in C
\bigm\| \bigm\| f \prime \prime (z)
\bigm\| \bigm\|
\scrL (X2;Y )
\| x - y\| 2X
for any x, y \in C.
The constants
1
12
and
1
24
are best possible.
5. Related inequalities. We have the following result as well:
Theorem 5.1. Let (X; \| \cdot \| X) and (Y ; \| \cdot \| Y ) be two normed linear spaces over the complex
number field \BbbC with Y complete. Assume that the mapping F : C \subset X \rightarrow Y is continuous on the
convex set C in the norm topology. If F \in \scrB \scrN K(C) for some K > 0, then we have\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
F (uy + (1 - u)x)du - 1
2\lambda - 1
\lambda \int
1 - \lambda
F (sx+ (1 - s)y)ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
F
\leq
\leq 1
6
K\lambda (1 - \lambda )\| y - x\| 2X (5.1)
for any \lambda \in [0, 1], \lambda \not = 1
2
and x, y \in C.
Proof. Since F \in \scrB \scrN K(C) for K > 0, then
\| (1 - \lambda )F (u) + \lambda F (v) - F ((1 - \lambda )u+ \lambda v)\| Y \leq 1
2
K\lambda (1 - \lambda )\| u - v\| 2X (5.2)
for any u, v \in C and \lambda \in [0, 1].
Let t \in [0, 1] and for x, y \in C, take
u = (1 - t)((1 - \lambda )x+ \lambda y) + ty, v = tx+ (1 - t)((1 - \lambda )x+ \lambda y) \in C
in (5.2) to get
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INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1343
\| (1 - \lambda )F ((1 - t)((1 - \lambda )x+ \lambda y) + ty) + \lambda F (tx+ (1 - t)((1 - \lambda )x+ \lambda y)t) -
- F ((1 - \lambda )[(1 - t)((1 - \lambda )x+ \lambda y) + ty] + \lambda [tx+ (1 - t)((1 - \lambda )x+ \lambda y)])\| Y \leq
\leq 1
2
K\lambda (1 - \lambda )\| (1 - t)((1 - \lambda )x+ \lambda y) + ty - [tx+ (1 - t)((1 - \lambda )x+ \lambda y)]\| 2X . (5.3)
Observe that
(1 - \lambda )[(1 - t)((1 - \lambda )x+ \lambda y) + ty] + \lambda [tx+ (1 - t)((1 - \lambda )x+ \lambda y)] =
= (1 - \lambda )(1 - t)((1 - \lambda )x+ \lambda y) + (1 - \lambda )ty + \lambda tx+ \lambda (1 - t)((1 - \lambda )x+ \lambda y) =
= (1 - t)((1 - \lambda )x+ \lambda y) + (1 - \lambda )ty + \lambda tx =
= [(1 - t)(1 - \lambda ) + \lambda t]x+ [(1 - t)\lambda + (1 - \lambda )t]y
and
(1 - t)((1 - \lambda )x+ \lambda y) + ty - [tx+ (1 - t)((1 - \lambda )x+ \lambda y)] =
= (1 - t)(1 - \lambda )x+ (1 - t)\lambda y + ty - tx - (1 - t)(1 - \lambda )x - (1 - t)\lambda y = t(y - x).
Then by (5.3) we have
\| (1 - \lambda )F ((1 - t)((1 - \lambda )x+ \lambda y) + ty) + \lambda F (tx+ (1 - t)((1 - \lambda )x+ \lambda y)) -
- F ([(1 - t)(1 - \lambda ) + \lambda t]x+ [(1 - t)\lambda + (1 - \lambda )t]y)\| Y \leq 1
2
K \lambda (1 - \lambda )t2\| y - x\| 2X (5.4)
for any t, \lambda \in [0, 1] and x, y \in C.
Integrating the inequality (5.4) over t on [0, 1] and using the generalized triangle inequality for
norms and integrals, we get\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| (1 - \lambda )
1\int
0
F ((1 - t)((1 - \lambda )x+ \lambda y) + ty)dt+
+\lambda
1\int
0
F (tx+ (1 - t)((1 - \lambda )x+ \lambda y))dt -
-
1\int
0
F ([(1 - t)(1 - \lambda ) + \lambda t]x+ [(1 - t)\lambda + (1 - \lambda )t]y)dt
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Y
\leq
\leq 1
6
K\lambda (1 - \lambda )\| y - x\| 2X (5.5)
for any \lambda \in [0, 1] and x, y \in C.
Observe that
1\int
0
F [(1 - t)(\lambda y + (1 - \lambda )x) + ty]dt =
1\int
0
F [((1 - t)\lambda + t)y + (1 - t)(1 - \lambda )x]dt
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1344 S. S. DRAGOMIR
and
1\int
0
F (tx+ (1 - t)((1 - \lambda )x+ \lambda y))dt =
=
1\int
0
F ((1 - t)x+ t((1 - \lambda )x+ \lambda y))dt =
1\int
0
F [t\lambda y + (1 - \lambda t)x]dt.
If we make the change of variable u := (1 - t)\lambda + t, then we have 1 - u = (1 - t)(1 - \lambda ) and
du = (1 - \lambda )du. Then
1\int
0
F [((1 - t)\lambda + t)y + (1 - t)(1 - \lambda )x]dt =
1
1 - \lambda
1\int
\lambda
F [uy + (1 - u)x]du.
If we make the change of variable u := \lambda t, then we have du = \lambda dt and
1\int
0
F [t\lambda y + (1 - \lambda t)x]dt =
1
\lambda
\lambda \int
0
F [uy + (1 - u)x]du.
Therefore
(1 - \lambda )
1\int
0
F [(1 - t)(\lambda y + (1 - \lambda )x) + ty]dt+
+\lambda
1\int
0
F [t(\lambda y + (1 - \lambda )x) + (1 - t)x]dt =
=
1\int
\lambda
F [uy + (1 - u)x]du+
\lambda \int
0
F [uy + (1 - u)x]du =
1\int
0
F [uy + (1 - u)x]du,
and we have the simple equality
(1 - \lambda )
1\int
0
F ((1 - t)((1 - \lambda )x+ \lambda y) + ty)dt+
+\lambda
1\int
0
F (tx+ (1 - t)((1 - \lambda )x+ \lambda y))dt =
1\int
0
F [uy + (1 - u)x]du
for any \lambda \in [0, 1] and x, y \in C.
Consider now the integral
1\int
0
F ([(1 - t)(1 - \lambda ) + \lambda t]x+ [(1 - t)\lambda + (1 - \lambda )t]y)dt.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1345
Put
s = (1 - t)(1 - \lambda ) + \lambda t = 1 - \lambda + (2\lambda - 1)t.
Then
1 - s = (1 - t)\lambda + (1 - \lambda )t.
If \lambda \not = 1
2
, then s = 1 - \lambda + (2\lambda - 1)t is a change of variable with dt =
1
2\lambda - 1
and we have
1\int
0
F ([(1 - t)(1 - \lambda ) + \lambda t]x+ [(1 - t)\lambda + (1 - \lambda )t]y)dt =
=
1
2\lambda - 1
\lambda \int
1 - \lambda
F (sx+ (1 - s)y)ds.
Now, making use of (5.5) we get the desired result (5.1).
Theorem 5.1 is proved.
Remark 5.1. We observe that for \lambda \rightarrow 1
2
we recapture from (5.1) the inequality (2.8).
If we take in (5.1) \lambda =
3
4
, then we get\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
F [uy + (1 - u)x]du - 2
3/4\int
1/4
F (sx+ (1 - s)y)ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
F
\leq 1
32
K\| y - x\| 2X .
Let f(z) =
\sum \infty
n=0
\alpha nz
n be a function defined by power series with complex coefficients and
convergent on the open disk D(0, R) \subset \BbbC , R > 0. For any x, y in the Banach algebra \scrB with
\| x\| , \| y\| \leq M < R, M > 0 we have\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
f(uy + (1 - u)x)du - 1
2\lambda - 1
\lambda \int
1 - \lambda
f(sx+ (1 - s)y)ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq 1
6
f \prime \prime
abs(M)\lambda (1 - \lambda )\| y - x\| 2
for any \lambda \in [0, 1], \lambda \not = 1
2
.
Let (X, \| \cdot \| X) and (Y, \| \cdot \| Y ) be two normed linear spaces, with Y complete, C an open convex
subset of X and f : C \rightarrow Y a twice-differentiable mapping on C. Then for any x, y \in C and
\lambda \in [0, 1], \lambda \not = 1
2
, we obtain\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1\int
0
f(uy + (1 - u)x)du - 1
2\lambda - 1
\lambda \int
1 - \lambda
f(sx+ (1 - s)y)ds
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \leq
\leq 1
6
\mathrm{s}\mathrm{u}\mathrm{p}
z\in C
\bigm\| \bigm\| f \prime \prime (z)
\bigm\| \bigm\|
\scrL (X2;Y )
\lambda (1 - \lambda )\| y - x\| 2.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1346 S. S. DRAGOMIR
References
1. Anastassiou G. A. Univariate Ostrowski inequalities, revisited // Monatsh. Math. – 2002. – 135, № 3. – S. 175 – 189.
2. Araki H., Yamagami S. An inequality for Hilbert – Schmidt norm // Communs Math. Phys. – 1981. – 81. – P. 89 – 96.
3. Bhatia R. First and second order perturbation bounds for the operator absolute value // Linear Algebra and Appl. –
1994. – 208/209. – P. 367 – 376.
4. Bhatia R. Perturbation bounds for the operator absolute value // Linear Algebra and Appl. – 1995. – 226/228. –
P. 639 – 645.
5. Bhatia R., Singh D., Sinha K. B. Differentiation of operator functions and perturbation bounds // Communs Math.
Phys. – 1998. – 191, № 3. – P. 603 – 611.
6. Bhatia R. Matrix analysis. – Springer-Verlag, 1997.
7. Cerone P., Dragomir S. S. Midpoint-type rules from an inequalities point of view // Handb. Anal.-Comput. Meth.
Appl. Math. /Ed. G. A. Anastassiou. – New York: CRC Press, 2003. – P. 135 – 200.
8. Cerone P., Dragomir S. S. New bounds for the three-point rule involving the Riemann – Stieltjes integrals // Adv.
Statist. Combin. and Relat. Areas. – World Sci. Publ., 2002. – P. 53 – 62.
9. Cerone P., Dragomir S. S., Roumeliotis J. Some Ostrowski type inequalities for n-time differentiable mappings and
applications // Demonstr. Math. – 1999. – 32, № 2. – P. 697 – 712.
10. Ciurdariu L. A note concerning several Hermite – Hadamard inequalities for different types of convex functions //
Int. J. Math. Anal. – 2012. – 6, № 33 – 36. – P. 1623 – 1639.
11. Coleman R. Calculus on normed vector spaces. – New York etc.: Springer, 2012.
12. Dragomir S. S., Cho Y. J., Kim S. S. Inequalities of Hadamard’s type for Lipschitzian mappings and their applications
// J. Math. Anal. and Appl. – 2000. – 245, № 2. – P. 489 – 501.
13. Dragomir S. S., Pearce C. E. M. Selected topics on Hermite – Hadamard inequalities and applications // RGMIA
Monogr. – 2000 [Online http://rgmia.org/monographs/hermite_hadamard.html].
14. Dragomir S. S. Semi-inner products and applications. – Hauppauge, NY: Nova Sci. Publ., Inc. – 2004. – x+222 p.
15. Dragomir S. S. Ostrowski’s inequality for monotonous mappings and applications // J. KSIAM. – 1999. – 3, № 1. –
P. 127 – 135.
16. Dragomir S. S. The Ostrowski’s integral inequality for Lipschitzian mappings and applications // Comput. Math. and
Appl. – 1999. – 38. – P. 33 – 37.
17. Dragomir S. S. On the Ostrowski’s inequality for Riemann – Stieltjes integral // Korean J. Appl. Math. – 2000. – 7. –
P. 477 – 485.
18. Dragomir S. S. On the Ostrowski’s inequality for mappings of bounded variation and applications // Math. Inequal.
and Appl. – 2001. – 4, № 1. – P. 33 – 40.
19. Dragomir S. S. On the Ostrowski inequality for Riemann – Stieltjes integral
\int b
a
f(t)du(t) where f is of Hölder type
and u is of bounded variation and applications // J. KSIAM. – 2001. – 5, № 1. – P. 35 – 45.
20. Dragomir S. S. Ostrowski type inequalities for isotonic linear functionals // J. Inequal. Pure and Appl. Math. – 2002. –
3, № 5. – Art. 68.
21. Dragomir S. S. An inequality improving the first Hermite – Hadamard inequality for convex functions defined on
linear spaces and applications for semi-inner products // J. Inequal. Pure and Appl. Math. – 2002. – 3, № 2. – Art. 31. –
8 p.
22. Dragomir S. S. An Ostrowski like inequality for convex functions and applications // Rev. Math. Comput. – 2003. –
16, № 2. – P. 373 – 382.
23. Dragomir S. S. Operator inequalities of Ostrowski and trapezoidal type // Springer Briefs in Math. – New York:
Springer, 2012. – x+112 p.
24. Dragomir S. S. Inequalities for power series in Banach algebras // SUT J. Math. – 2014. – 50, № 1. – P. 25 – 45.
25. Dragomir S. S. Integral inequalities for Lipschitzian mappings between two Banach spaces and applications // Kodai
Math. J. – 2016. – 39. – P. 227 – 251.
26. Dragomir S. S., Cerone P., Roumeliotis J., Wang S. A weighted version of Ostrowski inequality for mappings of
Hölder type and applications in numerical analysis // Bull. math. Soc. sci. math. roum. – 1999. – 42(90), № 4. –
P. 301 – 314.
27. Dragomir S. S., Rassias Th. M. (Eds). Ostrowski type inequalities and applications in numerical integration. – Kluwer
Acad. Publ., 2002.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
INTEGRAL INEQUALITIES OF THE HERMITE – HADAMARD . . . 1347
28. Dragomir S. S., Wang S. A new inequality of Ostrowski’s type in L1 -norm and applications to some special means
and to some numerical quadrature rules // Tamkang J. Math. – 1997. – 28. – P. 239 – 244.
29. Dragomir S. S., Wang S. Applications of Ostrowski’s inequality to the estimation of error bounds for some special
means and some numerical quadrature rules // Appl. Math. Lett. – 1998. – 11. – P. 105 – 109.
30. Dragomir S. S., Wang S. A new inequality of Ostrowski’s type in Lp -norm and applications to some special means
and to some numerical quadrature rules // Indian J. Math. – 1998. – 40, № 3. – P. 245 – 304.
31. Douglas R. Banach algebra techniques in operator theory. – Acad. Press, 1972.
32. Farforovskaya Yu. B. Estimates of the closeness of spectral decompositions of self-adjoint operators in the Kan-
torovich – Rubinshtein metric (in Russian) // Vestn. Leningrad. Gos. Univ. Ser. Mat., Mekh., Astron. – 1967. – 4. –
P. 155 – 156.
33. Farforovskaya Yu. B. An estimate of the \mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m} \| f(B) - f(A)\| for self-adjoint operators A and B (in Russian) //
Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. – 1976. – 56. – P. 143 – 162.
34. Farforovskaya Yu. B., Nikolskaya L. Modulus of continuity of operator functions // Algebra i Analiz. – 2008. – 20,
№ 3. – P. 224 – 242 (transl. in St. Petersburg Math. J. – 2009. – 20, № 3. – P. 493 – 506).
35. Feng Y., Zhao W. Refinement of Hermite – Hadamard inequality // Far East J. Math. Sci. – 2012. – 68, № 2. –
P. 245 – 250.
36. Fink A. M. Bounds on the deviation of a function from its averages // Czechoslovak Math. J. – 1992. – 42(117),
№ 2. – P. 298 – 310.
37. Gao X. A note on the Hermite – Hadamard inequality // J. Math. Inequal. – 2010. – 4, № 4. – P. 587 – 591.
38. Hwang S.-R., Tseng K.-L., Hsu K.-C. Hermite – Hadamard type and Fejér type inequalities for general weights (I) //
J. Inequal. Appl. – 2013. – 2013, № 170.
39. Kato T. Continuity of the map S \rightarrow | S| for linear operators // Proc. Jap. Acad. – 1973. – 49. – P. 143 – 162.
40. Kırmacı U. S., Dikici R. On some Hermite – Hadamard type inequalities for twice differentiable mappings and
applications // Tamkang J. Math. – 2013. – 44, № 1. – P. 41 – 51.
41. Mikusiński J. The Bochner Integral. – Birkhäuser-Verlag, 1978.
42. Muddassar M., Bhatti M. I., Iqbal M. Some new s-Hermite – Hadamard type inequalities for differentiable functions
and their applications // Proc. Pakistan Acad. Sci. – 2012. – 49, № 1. – P. 9 – 17.
43. Matić M., Pečarić J. Note on inequalities of Hadamard’s type for Lipschitzian mappings // Tamkang J. Math. –
2001. – 32, № 2. – P. 127 – 130.
44. Ostrowski A. Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert // Com-
ment. math. helv. – 1938. – 10. – P. 226 – 227.
45. Rudin W. Functional analysis. – McGraw Hill, 1973.
46. Sarikaya M. Z. On new Hermite – Hadamard Fejér type integral inequalities // Stud. Univ. Babeş-Bolyai Math. –
2012. – 57, № 3. – P. 377 – 386.
47. Wąsowicz S., Witkowski A. On some inequality of Hermite – Hadamard type // Opusc. Math. – 2012. – 32, № 3. –
P. 591 – 600.
48. Xi B.-Y., Qi F. Some integral inequalities of Hermite – Hadamard type for convex functions with applications to means
// J. Funct. Spaces Appl. – 2012. – Art. ID 980438. – 14 p.
49. Zabandan G., Bodaghi A., Kılıçman A. The Hermite – Hadamard inequality for r-convex functions // J. Inequal.
Appl. – 2012. – 2012, № 215. – 8 p.
50. Zhao C. -J., Cheung W. -S., Li X.-Y. On the Hermite – Hadamard type inequalities // J. Inequal. Appl. – 2013. – 2013,
№ 228.
Received 06.10.15
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|
| id | umjimathkievua-article-1924 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T02:15:19Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/75/2c9650f0ca39565e96cf1adb75f77275.pdf |
| spelling | umjimathkievua-article-19242019-12-05T09:31:57Z Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex mappings Інтегральнi нерiвностi типу Ермiта–Адамара для $K$ -обмежених вiдображень з опуклою нормою Dragomir, S. S. Драгомир, С. С. Драгомир, С. С. We obtain some inequalities of the Hermite – Hadamard type for $K$-bounded norm convex mappings between two normed spaces. The applications for twice differentiable functions in Banach spaces and functions defined by power series in Banach algebras are presented. Some discrete Jensen-type inequalities are also obtained. Отримано деякi нерiвностi типу Ермiта – Адамара для $K$-обмежених вiдображень з опуклою нормою мiж двома нормованими просторами. Наведено застосування до двiчi диференцiйовних функцiй у банахових просторах та функцiй, що визначенi степеневими рядами в банахових алгебрах. Отримано також деякi нерiвностi типу Джексона. Institute of Mathematics, NAS of Ukraine 2016-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1924 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 10 (2016); 1330-1347 Український математичний журнал; Том 68 № 10 (2016); 1330-1347 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/1924/906 Copyright (c) 2016 Dragomir S. S. |
| spellingShingle | Dragomir, S. S. Драгомир, С. С. Драгомир, С. С. Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex mappings |
| title | Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex
mappings |
| title_alt | Інтегральнi нерiвностi типу Ермiта–Адамара для $K$ -обмежених вiдображень з опуклою нормою |
| title_full | Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex
mappings |
| title_fullStr | Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex
mappings |
| title_full_unstemmed | Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex
mappings |
| title_short | Integral inequalities of the Hermite – Hadamard type for $K$ -bounded norm convex
mappings |
| title_sort | integral inequalities of the hermite – hadamard type for $k$ -bounded norm convex
mappings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1924 |
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