On the generalization of some Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals
We provide a unified approach to getting Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via the Riemann – Liouville integrals. Some particular inequalities generalizing the classical results, such as the trapezoid inequality, Simpson’s inequality,...
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2018
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507495921876992 |
|---|---|
| author | Chen, F. X. Чен, Ф. Х. |
| author_facet | Chen, F. X. Чен, Ф. Х. |
| author_sort | Chen, F. X. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:22:46Z |
| description | We provide a unified approach to getting Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via the Riemann – Liouville integrals. Some particular inequalities generalizing the classical results, such as the trapezoid inequality, Simpson’s inequality, and midpoint inequality are also presented. |
| first_indexed | 2026-03-24T02:10:14Z |
| format | Article |
| fulltext |
UDC 517.5
F. X. Chen (Key Laboratory for Nonlinear Sci. and System Structure, School Math. and Statistics, Chongqing Three
Gorges Univ., China)
ON THE GENERALIZATION OF SOME HERMITE – HADAMARD
INEQUALITIES FOR FUNCTIONS WITH CONVEX ABSOLUTE VALUES
OF THE SECOND DERIVATIVES VIA FRACTIONAL INTEGRALS*
ПРО УЗАГАЛЬНЕННЯ ДЕЯКИХ НЕРIВНОСТЕЙ ЕРМIТА – АДАМАРА
ДЛЯ ФУНКЦIЙ З ОПУКЛИМИ АБСОЛЮТНИМИ ЗНАЧЕННЯМИ ДРУГИХ
ПОХIДНИХ ЗА ДОПОМОГОЮ IНТЕГРАЛIВ ДРОБОВОГО ПОРЯДКУ
We provide a unified approach to getting Hermite – Hadamard inequalities for functions with convex absolute values of
the second derivatives via the Riemann – Liouville integrals. Some particular inequalities generalizing the classical results,
such as the trapezoid inequality, Simpson’s inequality, and midpoint inequality are also presented.
Запропоновано унiфiкований пiдхiд до отримання нерiвностей Ермiта – Адамара для функцiй з опуклими абсолют-
ними значеннями других похiдних за допомогою iнтегралiв Рiмана – Лiувiлля. Наведено деякi частиннi нерiвностi,
що узагальнюють класичнi результати, такi як нерiвнiсть трапецiй, нерiвнiсть Сiмпсона та нерiвнiсть середньої
точки.
1. Introduction. Let f : I \subset R \rightarrow R be a convex function on the interval I, then for any a, b \in I
with a \not = b we have the following double inequality:
f
\biggl(
a+ b
2
\biggr)
\leq 1
b - a
b\int
a
f(t)dt \leq f(a) + f(b)
2
. (1)
This remarkable result is well known in the literature as the Hermite – Hadamard inequality. Note
that some of the the classical inequalities for means can be derived from (1) for appropriate parti-
cular selections of the mapping f. Both inequalities hold in the reversed direction if f is concave.
Some refinements of the Hermite – Hadamard inequality on convex functions have been extensively
investigated by a number of authors (see [2 – 7, 12]).
In [8], M. Z. Sarikaya et al. established some inequalities for twice differentiable convex map-
pings which are connected with Hadamard’s inequality, and they used the following lemma to prove
their results.
Lemma 1.1. Let f : I \subseteq \BbbR \rightarrow R be a twice differentiable mapping on I0, a, b \in I0 with a < b.
If f \in L1[a, b], then
1
b - a
b\int
a
f(x)dx - f
\biggl(
a+ b
2
\biggr)
=
(b - a)2
2
1\int
0
s(t)[f \prime \prime (ta+ (1 - t)b) + f \prime \prime (tb+ (1 - t)a)]dt,
where
* This work is supported by Innovation Team Building at Institutions of Higher Education in Chongqing
(CXTDX201601035) and the Science and Technology Research Program of Chongqing Municipal Education Commi-
ssion (Grant No. KJQN201801205).
c\bigcirc F. X. CHEN, 2018
1696 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON THE GENERALIZATION OF SOME HERMITE – HADAMARD INEQUALITIES . . . 1697
s(t) :=
\left\{
t2, t \in
\biggl[
0,
1
2
\biggr]
,
(1 - t)2, t \in
\biggl(
1
2
, 1
\biggr]
.
In [1], Alomari et al. obtained some inequalities for functions with quasiconvex absolute values of the
second derivatives connecting with the Hermite – Hadamard inequality on the basis of the following
lemma.
Lemma 1.2. Let f : I \subseteq \BbbR \rightarrow R be a twice differentiable mapping on I0, a, b \in I0 with a < b.
If f \in L1[a, b], then
f(a) + f(b)
2
- 1
b - a
b\int
a
f(x)dx =
(b - a)2
2
1\int
0
t(1 - t)f \prime \prime (ta+ (1 - t)b)dt.
The following general integral identity for functions with convex absolute values of the second
derivatives is proposed by M. Z. Sarikaya in [9].
Lemma 1.3. Let I \subseteq \BbbR be an open interval, a, b \in I with a < b. If f : I \rightarrow R is a twice
differentiable mapping such that f \prime \prime is integrable and 0 \leq \lambda \leq 1, then the following equality holds:
(\lambda - 1)f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
+
1
b - a
b\int
a
f(x)dx =
=
(b - a)2
2
1\int
0
Q(t)f \prime \prime (ta+ (1 - t)b)dt,
where
Q(t) :=
\left\{
t(t - \lambda ), t \in
\biggl[
0,
1
2
\biggr]
,
(1 - t)(1 - \lambda - t), t \in
\biggl(
1
2
, 1
\biggr]
.
It is remarkable that M. Z. Sarikaya et al. [10] proved the following interesting inequalities of
Hermite – Hadamard type involving Riemann – Liouville fractional integrals.
Theorem 1.1. Let f : [a, b] \rightarrow R be a positive function with a < b and f \in L1[a, b]. If f is a
convex function on [a, b], then the following inequalities for fractional integrals hold:
f
\biggl(
a+ b
2
\biggr)
\leq \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] \leq
f(a) + f(b)
2
(2)
with \alpha > 0.
We remark that the symbol J\alpha
a+ and J\alpha
b - f denote the left-hand and right-hand Riemann –
Liouville fractional integrals of the order \alpha \geq 0 with a \geq 0 which are defined by
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1698 F. X. CHEN
J\alpha
a+f(x) =
1
\Gamma (\alpha )
x\int
a
(x - t)\alpha - 1f(t)dt, x > a,
and
J\alpha
b - f(x) =
1
\Gamma (\alpha )
b\int
x
(t - x)\alpha - 1f(t)dt, x < b,
respectively. Here \Gamma (\alpha ) is the Gamma function defined by \Gamma (\alpha ) =
\int \infty
0
e - tt\alpha - 1dt.
J. R. Wang et al. [11] established the following fundamental integral identity including the second
order derivatives of a given function via Riemann – Liouville integrals.
Lemma 1.4. Let f : [a, b] \rightarrow R be a twice differentiable mapping on (a, b) with a < b. If
f \in L1[a, b], then the following equality for fractional integrals holds:
\Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] -
f(a) + f(b)
2
=
=
(b - a)2
2
1\int
0
(1 - t)\alpha +1 + t\alpha +1 - 1
\alpha + 1
f \prime \prime (ta+ (1 - t)b)dt. (3)
Another integral identity including the second order derivatives of a given function via Riemann –
Liouville integrals is obtained by Y. R. Zhang and J. R. Wang in [13] as follows.
Lemma 1.5. Let f : [a, b] \rightarrow R be a twice differentiable mapping on (a, b) with a < b. If
f \in L1[a, b], then
\Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - f
\biggl(
a+ b
2
\biggr)
=
(b - a)2
2
1\int
0
m(t)f \prime \prime (ta+ (1 - t)b)dt, (4)
where
m(t) :=
\left\{
t - 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
, t \in
\biggl[
0,
1
2
\biggr]
,
1 - t - 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
, t \in
\biggl(
1
2
, 1
\biggr]
.
In this paper, we generalize the results (3) and (4) for functions with convex absolute values of
the second derivatives via Riemann – Liouville integrals.
2. Main results. In order to prove our main theorems, we need the following lemma.
Lemma 2.1. Let f : [a, b] \rightarrow R be a twice differentiable mapping on (a, b) with a < b such that
f \prime \prime is integrable and 0 \leq \lambda \leq 1, then the following equality for fractional integrals holds:
\Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
=
=
(b - a)2
2
1\int
0
k(t)f \prime \prime (ta+ (1 - t)b)dt, (5)
where
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON THE GENERALIZATION OF SOME HERMITE – HADAMARD INEQUALITIES . . . 1699
k(t) :=
\left\{
t(1 - \lambda ) - 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
, t \in
\biggl[
0,
1
2
\biggr]
,
(1 - t)(1 - \lambda ) - 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
, t \in
\biggl(
1
2
, 1
\biggr]
.
Proof. Multiplying (3) by \lambda , (4) by 1 - \lambda on both sides, respectively, and adding the resulting
inequalities, we get (5). Then we get the desired result.
By using this lemma, we can obtain the following general integral inequalities.
Theorem 2.1. Let f : [a, b] \rightarrow R be a twice differentiable mapping on (a, b) with a < b such
that f \prime \prime is integrable and 0 \leq \lambda \leq 1. If | f \prime \prime | is convex on [a, b], then the following inequality holds:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq
\leq (b - a)2
2
\biggl[
\alpha
2(\alpha + 1)(\alpha + 2)
+
1 - \lambda
8
\biggr]
(| f \prime \prime (a)| + | f \prime \prime (b)| ).
Proof. From Lemma 2.1 and the definition of k(t), we have
\Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\leq
\leq (b - a)2
2
\left\{
1
2\int
0
\bigm| \bigm| \bigm| \bigm| t(1 - \lambda ) - 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\bigm| \bigm| \bigm| \bigm| | f \prime \prime (ta+ (1 - t)b)| dt+
+
1\int
1
2
\bigm| \bigm| \bigm| \bigm| (1 - t)(1 - \lambda ) - 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\bigm| \bigm| \bigm| \bigm| | f \prime \prime (ta+ (1 - t)b)| dt
\right\} \leq
\leq (b - a)2
2
\left\{
1
2\int
0
t(1 - \lambda )| f \prime \prime (ta+ (1 - t)b)| dt+
1\int
1
2
(1 - t)(1 - \lambda )| f \prime \prime (ta+ (1 - t)b)| dt+
+
1\int
0
\bigm| \bigm| \bigm| \bigm| 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\bigm| \bigm| \bigm| \bigm| | f \prime \prime (ta+ (1 - t)b)| dt
\right\} . (6)
Because (1 - t)\alpha +1 + t\alpha +1 \leq 1 for any t \in [0, 1] and | f \prime \prime | is convex on [a, b], we get
1\int
0
\bigm| \bigm| \bigm| \bigm| 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\bigm| \bigm| \bigm| \bigm| | f \prime \prime (ta+ (1 - t)b)| dt \leq
\leq
1\int
0
1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\bigl(
t| f \prime \prime (a)| + (1 - t)| f \prime \prime (b)|
\bigr)
dt =
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1700 F. X. CHEN
=
\alpha
(\alpha + 1)(\alpha + 2)
| f \prime \prime (a)| + | f \prime \prime (b)|
2
. (7)
On the other hand,
1
2\int
0
t(1 - \lambda )| f \prime \prime (ta+ (1 - t)b)| dt+
1\int
1
2
(1 - t)(1 - \lambda )| f \prime \prime (ta+ (1 - t)b)| dt =
=
(1 - \lambda )(| f \prime \prime (a)| + | f \prime \prime (b)| )
8
. (8)
Now by (6) – (8), we can obtain the desired result which completes the proof.
Theorem 2.2. Let f : [a, b] \rightarrow R be a twice differentiable mapping on (a, b) with a < b such
that | f \prime \prime | is integrable and 0 \leq \lambda \leq 1. If | f \prime \prime | q is convex on [a, b] with q > 1, then the following
inequality holds:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq
\leq (b - a)2(1 - \lambda )
2
\biggl(
2
p+ 1
\biggr) 1
p
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
\Biggr] 1
p
\times
\times
\biggl(
| f \prime \prime (a)| q + | f \prime \prime (b)| q
2
\biggr) 1
q
,
where
1
p
+
1
q
= 1.
Proof. From Lemma 2.1 and the definition of k(t), by using the Hölder inequality, we have\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq
\leq (b - a)2
2
1\int
0
| k(t)| | f \prime \prime (ta+ (1 - t)b)| dt \leq
\leq (b - a)2
2
\left( 1\int
0
| k(t)| pdt
\right)
1
p
\left( 1\int
0
| f \prime \prime (ta+ (1 - t)b)| qdt
\right)
1
q
=
=
(b - a)2
2
\left(
1
2\int
0
| k1(t)| pdt+
1\int
1
2
| k2(t)| pdt
\right)
1
p \left( 1\int
0
| f \prime \prime (ta+ (1 - t)b)| qdt
\right)
1
q
, (9)
where
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON THE GENERALIZATION OF SOME HERMITE – HADAMARD INEQUALITIES . . . 1701
k1(t) = t(1 - \lambda ) - 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
,
k2(t) = (1 - t)(1 - \lambda ) - 1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
.
Because (1 - t)\alpha +1 + t\alpha +1 \leq 1 for any t \in [0, 1], using one skill of shrinking about inequality, we
get
1
2\int
0
| k1(t)| pdt \leq
1
2\int
0
\biggl(
t(1 - \lambda ) +
1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\biggr) p
dt \leq
\leq
1
2\int
0
\biggl(
t(1 - \lambda ) +
1
\alpha + 1
\biggr) p
dt = (1 - \lambda )p
1
2\int
0
\biggl(
t+
1
(\alpha + 1)(1 - \lambda )
\biggr) p
dt,
1
2\int
0
\biggl(
t+
1
(\alpha + 1)(1 - \lambda )
\biggr) p
dt =
=
1
p+ 1
\biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
- 1
p+ 1
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
,
and
1\int
1
2
| k2(t)| pdt \leq
1\int
1
2
\biggl(
(1 - t)(1 - \lambda ) +
1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\biggr) p
dt \leq
\leq
1\int
1
2
\biggl(
(1 - t)(1 - \lambda ) +
1
\alpha + 1
\biggr) p
dt = (1 - \lambda )p
1\int
1
2
\biggl(
(1 - t) +
1
(\alpha + 1)(1 - \lambda )
\biggr) p
dt,
1\int
1
2
\biggl(
(1 - t) +
1
(\alpha + 1)(1 - \lambda )
\biggr) p
dt =
=
1
p+ 1
\biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
- 1
p+ 1
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
.
Thus,
1\int
0
| k(t)| pdt = (1 - \lambda )p
2
p+ 1
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
\Biggr]
. (10)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1702 F. X. CHEN
Moreover, because | f \prime \prime | q is convex on [a, b], we obtain
1\int
0
| f \prime \prime (ta+ (1 - t)b)| qdt \leq | f \prime \prime (a)| q + | f \prime \prime (b)| q
2
. (11)
Thus, submitting (9) and (10) to (11), we can derive the desired result.
Corollary 2.1. With the assumptions as in Theorem 2.2, if | f \prime \prime (x)| \leq M on [a, b], we have the
following inequality:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq
\leq M(b - a)2(1 - \lambda )
2
\biggl(
2
p+ 1
\biggr) 1
p
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
\Biggr] 1
p
.
Another Hermite – Hadamard inequalities for powers in terms of the second derivatives are given
as follows.
Theorem 2.3. Let f : [a, b] \rightarrow R be a twice differentiable mapping on (a, b) with a < b such
that | f \prime \prime | is integrable and 0 \leq \lambda \leq 1. If | f \prime \prime | q is convex on [a, b] with q > 1, then the following
inequality holds:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq
\leq (b - a)2(1 - \lambda )
2
\biggl(
2
q + 1
\biggr) 1
q
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
\Biggr] 1
q
\times
\times
\biggl(
| f \prime \prime (a)| q + | f \prime \prime (b)| q
2
\biggr) 1
q
.
Proof. From Lemma 2.1 and using the Hölder inequality, we have\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq
\leq (b - a)2
2
\left( 1\int
0
1dt
\right)
1
p
\left( 1\int
0
| k(t)f \prime \prime (ta+ (1 - t)b)| qdt
\right)
1
q
\leq
\leq (b - a)2
2
\left( | f \prime \prime (a)| q
1\int
0
t| k(t)| qdt+ | f \prime \prime (b)| q
1\int
0
(1 - t)| k(t)| qdt
\right)
1
q
. (12)
Calculating by parts, we get
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON THE GENERALIZATION OF SOME HERMITE – HADAMARD INEQUALITIES . . . 1703
1\int
0
t| k(t)| qdt =
1
2\int
0
t| k1(t)| qdt+
1\int
1
2
t| k2(t)| qdt
with
1
2\int
0
t| k1(t)| qdt \leq
1
2\int
0
t
\biggl(
t(1 - \lambda ) +
1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\biggr) q
dt \leq
\leq
1
2\int
0
t
\biggl(
t(1 - \lambda ) +
1
\alpha + 1
\biggr) q
dt = (1 - \lambda )q
1
2\int
0
t
\biggl(
t+
1
(\alpha + 1)(1 - \lambda )
\biggr) q
dt
and
1\int
1
2
t| k2(t)| qdt \leq
1\int
1
2
t
\biggl(
(1 - t)(1 - \lambda ) +
1 - (1 - t)\alpha +1 - t\alpha +1
\alpha + 1
\biggr) q
dt \leq
\leq
1\int
1
2
t
\biggl(
(1 - t)(1 - \lambda ) +
1
\alpha + 1
\biggr) q
dt = (1 - \lambda )q
1\int
1
2
t
\biggl(
(1 - t) +
1
(\alpha + 1)(1 - \lambda )
\biggr) q
dt,
where
1
2\int
0
t
\biggl(
t+
1
(\alpha + 1)(1 - \lambda )
\biggr) q
dt =
1
2(q + 1)
\biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
-
- 1
(q + 1)(q + 2)
\biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+2
+
1
(q + 1)(q + 2)
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) q+2
and
1\int
1
2
t
\biggl(
(1 - t) +
1
(\alpha + 1)(1 - \lambda )
\biggr) q
dt = - 1
(q + 1)
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
+
+
1
2(q + 1)
\biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
- 1
(q + 1)(q + 2)
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) q+2
+
+
1
(q + 1)(q + 2)
\biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+2
.
Thus,
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1704 F. X. CHEN
1\int
0
t| k(t)| qdt \leq (1 - \lambda )q
1
q + 1
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
\Biggr]
. (13)
Similarly,
1\int
0
(1 - t)| k(t)| qdt \leq (1 - \lambda )q
1
q + 1
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
\Biggr]
. (14)
Thus, using (12) – (14), we get the desired result.
Corollary 2.2. With the assumptions as in Theorem 2.3, if | f \prime \prime (x)| \leq M on [a, b], we have the
following inequality:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq
\leq M(b - a)2(1 - \lambda )
2
\biggl(
2
q + 1
\biggr) 1
q
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
\Biggr] 1
q
.
Corollary 2.3. With the assumptions as in Theorems 2.2 and 2.3, we have\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq \mathrm{m}\mathrm{i}\mathrm{n}\{ N1, N2\} ,
where
N1 =
(b - a)2(1 - \lambda )
2
\biggl(
2
p+ 1
\biggr) 1
p
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
\Biggr] 1
p
\times
\times
\biggl(
| f \prime \prime (a)| q + | f \prime \prime (b)| q
2
\biggr) 1
q
and
N2 =
(b - a)2(1 - \lambda )
2
\biggl(
2
q + 1
\biggr) 1
q
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) q+1
\Biggr] 1
q
\times
\times
\biggl(
| f \prime \prime (a)| q + | f \prime \prime (b)| q
2
\biggr) 1
q
.
Theorem 2.4. Let f : [a, b] \rightarrow R be a twice differentiable mapping on (a, b) with a < b such
that | f \prime \prime | is integrable and 0 \leq \lambda \leq 1. If | f \prime \prime | q is concave on [a, b] with q > 1, then the following
inequality holds:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - (1 - \lambda )f
\biggl(
a+ b
2
\biggr)
- \lambda
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
ON THE GENERALIZATION OF SOME HERMITE – HADAMARD INEQUALITIES . . . 1705
\leq (b - a)2(1 - \lambda )
2
\biggl(
2
p+ 1
\biggr) 1
p
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
-
\biggl(
1
(\alpha + 1)(1 - \lambda )
\biggr) p+1
\Biggr] 1
p
\times
\times
\bigm| \bigm| \bigm| \bigm| f \prime \prime
\biggl(
a+ b
2
\biggr) \bigm| \bigm| \bigm| \bigm| ,
where
1
p
+
1
q
= 1.
Proof. Similarly as in Theorem 2.2, but now | f \prime \prime | q is concave on [a, b], we have
1\int
0
| f \prime \prime (ta+ (1 - t)b)| qdt \leq
\bigm| \bigm| \bigm| \bigm| f \prime \prime
\biggl(
a+ b
2
\biggr) \bigm| \bigm| \bigm| \bigm| q ,
so the desired result immediately follows.
3. Applications to quadrature formulas. In this section, we point out some particular inequa-
lities generalizing the classical results, such as the trapezoid inequality, Simpson’s inequality, and
midpoint inequality.
Proposition 3.1 (trapezoid inequality). Under the assumptions in Theorem 2.1 with \lambda = 1, we
get \bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] -
f(a) + f(b)
2
\bigm| \bigm| \bigm| \bigm| \leq \alpha (b - a)2
4(\alpha + 1)(\alpha + 2)
(| f \prime \prime (a)| + | f \prime \prime (b)| ).
Proposition 3.2 (midpoint inequality). Under the assumptions in Theorem 2.1 with \lambda = 0, we
have \bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - f
\biggl(
a+ b
2
\biggr) \bigm| \bigm| \bigm| \bigm| \leq
\leq (b - a)2
2
\biggl[
\alpha
2(\alpha + 1)(\alpha + 2)
+
1
8
\biggr]
(| f \prime \prime (a)| + | f \prime \prime (b)| ).
Proposition 3.3 (Simpson’s inequality). Under the assumptions in Theorem 2.1 with \lambda =
1
3
, we
obtain \bigm| \bigm| \bigm| \bigm| 16
\biggl[
f(a) + 4f
\biggl(
a+ b
2
\biggr)
+ f(b) - \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)]
\biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq (b - a)2
2
\biggl[
\alpha
2(\alpha + 1)(\alpha + 2)
+
1
12
\biggr]
(| f \prime \prime (a)| + | f \prime \prime (b)| ).
Proposition 3.4 (midpoint inequality). Under the assumptions in Theorem 2.2 with \lambda = 0, we
get \bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] - f
\biggl(
a+ b
2
\biggr) \bigm| \bigm| \bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
1706 F. X. CHEN
\leq (b - a)2
2
\biggl(
2
p+ 1
\biggr) 1
p
\Biggl[ \biggl(
1
2
+
1
(\alpha + 1)
\biggr) p+1
-
\biggl(
1
(\alpha + 1)
\biggr) p+1
\Biggr] 1
p
\times
\times
\biggl(
| f \prime \prime (a)| q + | f \prime \prime (b)| q
2
\biggr) 1
q
.
Proposition 3.5 (Simpson’s inequality). Under the assumptions in Theorem 2.3 with \lambda =
1
3
, we
have \bigm| \bigm| \bigm| \bigm| 16
\biggl[
f(a) + 4f
\biggl(
a+ b
2
\biggr)
+ f(b) - \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)]
\biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq (b - a)2
3
\biggl(
2
q + 1
\biggr) 1
q
\Biggl[ \biggl(
1
2
+
3
2(\alpha + 1)
\biggr) q+1
-
\biggl(
3
2(\alpha + 1)
\biggr) q+1
\Biggr] 1
q
\times
\times
\biggl(
| f \prime \prime (a)| q + | f \prime \prime (b)| q
2
\biggr) 1
q
.
References
1. Alomari M., Darus M., Dragomir S. S. New inequalities of Hermite – Hadamard type for functions whose second
derivatives absolute values are quasiconvex // RGMIA Res. Rep. Coll. – 2009. – 12. – Supplement, Article 17.
2. Bessenyei M. P\'ales Z. Hadamard-type inequalities for generalized convex functions // Math. Inequal. Appl. – 2003. –
6, №3. – P. 379 – 392.
3. Chu Y. M., Wang G. D., Zhang X. H. Schur convexity and Hadamard’s inequality // Math. Inequal. Appl. – 2010. –
13, № 4. – P. 725 – 731.
4. Farissi A. E. Simple proof and refinement of Hermite – Hadamard inequality // J. Math. Inequal. – 2010. – 4, № 3. –
P. 365 – 369.
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6. Gill P. M., Pearce C. E. M., Pe\v cari\'c J. Hadamard’s inequality for r-convex functions // J. Math. Anal. and Appl. –
1997. – 215, № 2. – P. 461 – 470.
7. Kirmaci U. S., Bakula M. K., \"Ozdemir M. E., Pe\v cari\'c J. Hadamard-type inequalities for s-convex functions // Appl.
Math. and Comput. – 2007. – 1(193). – P. 26 – 35.
8. Sarikaya M. Z. et al. New inequalities of Hermite – Hadamard type for functions whose second derivatives absolute
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Received 28.12.14,
after revision — 30.10.18
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 12
|
| id | umjimathkievua-article-1668 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:14Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/08/b4d3bb11dd189f0126026f678d769608.pdf |
| spelling | umjimathkievua-article-16682019-12-05T09:22:46Z On the generalization of some Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals Про узагальнення деяких нерiвностей Ермiта – Адамара для функцiй з опуклими абсолютними значеннями других похiдних за допомогою iнтегралiв дробового порядку Chen, F. X. Чен, Ф. Х. We provide a unified approach to getting Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via the Riemann – Liouville integrals. Some particular inequalities generalizing the classical results, such as the trapezoid inequality, Simpson’s inequality, and midpoint inequality are also presented. Запропоновано унiфiкований пiдхiд до отримання нерiвностей Ермiта – Адамара для функцiй з опуклими абсолютними значеннями других похiдних за допомогою iнтегралiв Рiмана – Лiувiлля. Наведено деякi частиннi нерiвностi, що узагальнюють класичнi результати, такi як нерiвнiсть трапецiй, нерiвнiсть Сiмпсона та нерiвнiсть середньої точки. Institute of Mathematics, NAS of Ukraine 2018-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1668 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 12 (2018); 1696-1706 Український математичний журнал; Том 70 № 12 (2018); 1696-1706 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1668/650 Copyright (c) 2018 Chen F. X. |
| spellingShingle | Chen, F. X. Чен, Ф. Х. On the generalization of some Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals |
| title | On the generalization of some Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals |
| title_alt | Про узагальнення деяких нерiвностей Ермiта – Адамара
для функцiй з опуклими абсолютними значеннями других
похiдних за допомогою iнтегралiв дробового порядку |
| title_full | On the generalization of some Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals |
| title_fullStr | On the generalization of some Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals |
| title_full_unstemmed | On the generalization of some Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals |
| title_short | On the generalization of some Hermite – Hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals |
| title_sort | on the generalization of some hermite – hadamard inequalities for functions with convex absolute values of the second derivatives via fractional integrals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1668 |
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