New fractional integral inequalities for differentiable convex functions and their applications
We establish some new fractional integral inequalities for differentiable convex functions and give several applications for the Beta-function.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507544355602432 |
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| author | Hsu, K.-C. Tseng, K.-L. Хсу, К.-С. Цзен, К.-Л. |
| author_facet | Hsu, K.-C. Tseng, K.-L. Хсу, К.-С. Цзен, К.-Л. |
| author_sort | Hsu, K.-C. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:24:16Z |
| description | We establish some new fractional integral inequalities for differentiable convex functions and give several applications for
the Beta-function. |
| first_indexed | 2026-03-24T02:11:00Z |
| format | Article |
| fulltext |
UDC 517.5
K.-L. Tseng, K.-C. Hsu (Aletheia Univ., Tamsui, Taiwan)
NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE
CONVEX FUNCTIONS AND THEIR APPLICATIONS
НОВI ДРОБОВО-IНТЕГРАЛЬНI НЕРIВНОСТI ДЛЯ ДИФЕРЕНЦIЙОВНИХ
ОПУКЛИХ ФУНКЦIЙ ТА ЇХ ЗАСТОСУВАННЯ
We establish some new fractional integral inequalities for differentiable convex functions and give several applications for
the Beta-function.
Встановлено деякi новi дробово-iнтегральнi нерiвностi для диференцiйовних опуклих функцiй i наведено кiлька
застосувань для бета-функцiї.
1. Introduction. Throughout in this paper, let a < b in \BbbR .
The inequality
f
\biggl(
a+ b
2
\biggr)
\leq 1
b - a
b\int
a
f(x) dx \leq f(a) + f(b)
2
(1.1)
which holds for all convex functions f : [a, b] \rightarrow \BbbR , is known in the literature as Hermite – Hadamard
inequality [7].
For some results which generalize, improve, and extend the inequality (1.1), see [1 – 6] and
[8 – 17].
In [14], Tseng et al. established the following Hermite – Hadamard-type inequality which refines
the inequality (1.1).
Theorem A. Suppose that f : [a, b] \rightarrow \BbbR is a convex function on [a, b]. Then we have the
inequality
f
\biggl(
a+ b
2
\biggr)
\leq 1
2
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr]
\leq
\leq 1
b - a
b\int
a
f(x) dx \leq
\leq 1
2
\biggl[
f
\biggl(
a+ b
2
\biggr)
+
(a) + f(b)
2
\biggr]
\leq f(a) + f(b)
2
. (1.2)
The third inequality in (1.2) is known in the literature as Bullen inequality.
In [4], Dragomir and Agarwal established the following results connected with the second in-
equality in the inequality (1.1).
Theorem B. Let f : [a, b] \rightarrow \BbbR be a differentiable function on (a, b) with a < b. If | f \prime | is
convex on [a, b], then we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f(a) + f(b)
2
- 1
b - a
b\int
a
f(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq b - a
8
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
which is the trapezoid inequality provided | f \prime | is convex on [a, b].
c\bigcirc K.-L. TSENG, K.-C. HSU, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3 407
408 K.-L. TSENG, K.-C. HSU
In [11], Kirmaci and Özdemir established the following results connected with the first inequality
in the inequality (1.1).
Theorem C. Under the assumptions of Theorem B, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
b - a
b\int
a
f(x) dx - f
\biggl(
a+ b
2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq b - a
8
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
which is the midpoint inequality provided | f \prime | is convex on [a, b].
In [12], Pearce and Pečarić established the following Hermite – Hadamard-type inequalities for
differentiable functions:
Theorem D. If f : Io \subseteq R \rightarrow R is a differentiable mapping on Io, a, b \in Io with a < b,
f \prime \in L1[a, b], q \geq 1 and | f \prime | q is convex on [a, b], then the following inequalities hold:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f(a) + f(b)
2
- 1
b - a
b\int
a
f(x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq b - a
4
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
,
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| 1
b - a
b\int
a
f(x) dx - f
\biggl(
a+ b
2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq b - a
4
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
.
In what follows we recall the following definition [13].
Definition 1.1. Let f \in L1[a, b]. The Riemann – Liouville integrals J\alpha
a+f and J\alpha
b - f of order
\alpha > 0 with a \geq 0 are defined by
J\alpha
a+f(x) =
1
\Gamma (\alpha )
x\int
a
(x - t)\alpha - 1 f(t) dt, x > a,
and
J\alpha
b - f(x) =
1
\Gamma (\alpha )
b\int
x
(t - x)\alpha - 1 f(t) dt, x < b,
respectively. Here \Gamma (\alpha ) is the Gamma-function and J0
a+f(x) = J0
b - f(x) = f(x).
In [13], Sarikaya et al. established the following Hermite – Hadamard-type inequalities for frac-
tional integrals:
Theorem E. Let f : [a, b] \rightarrow \BbbR be positive with 0 \leq a < b and f \in L1[a, b]. If f is a convex
function on [a, b], then
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
for \alpha > 0.
Theorem F. Under the assumptions of Theorem B, we have\bigm| \bigm| \bigm| \bigm| f(a) + f(b)
2
- \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)]
\bigm| \bigm| \bigm| \bigm| \leq
\leq 2\alpha - 1
2\alpha +1(\alpha + 1)
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
for \alpha > 0.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 409
In [9], Hwang et al. established the following fractional integral inequalities:
Theorem G. Under the assumptions of Theorem B, we have the following Hermite – Hadamard-
type inequality for fractional integrals:\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
4(\alpha + 1)
\biggl(
\alpha - 1 +
1
2\alpha - 1
\biggr) \bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr)
for \alpha > 0.
Theorem H. Under the assumptions of Theorem B, we have the following inequality for frac-
tional integrals with
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr)
2
:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] -
1
2
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl(
1
8
+
3\alpha +1 - 2\alpha +1 + 1
4\alpha +1(\alpha + 1)
- 1
2(\alpha + 1)
\biggr)
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) (1.3)
for \alpha > 0.
Theorem I. Under the assumptions of Theorem B, we have the following Bullen-type inequality
for fractional integrals: \bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] -
-
\biggl[
3\alpha - 1
4\alpha
f
\biggl(
a+ b
2
\biggr)
+
4\alpha - 3\alpha + 1
4\alpha
f(a) + f(b)
2
\biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
\alpha + 1
\biggl(
2\alpha + 1
2\alpha +1
- 3\alpha +1 + 1
4\alpha +1
\biggr)
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) (1.4)
for \alpha > 0.
Theorem J. Under the assumptions of Theorem B, we have the following Simpson-type inequality
for fractional integrals: \bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] -
-
\biggl[
5\alpha - 1
6\alpha
f
\biggl(
a+ b
2
\biggr)
+
6\alpha - 5\alpha + 1
6\alpha
f(a) + f(b)
2
\biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl[
1
\alpha + 1
\biggl(
2\alpha + 1
2\alpha +1
- 5\alpha +1 + 1
6\alpha +1
\biggr)
+
\biggl(
5\alpha - 1
12 \cdot 6\alpha
\biggr) \biggr]
(b - a)
\bigl( \bigm| \bigm| f \prime (a)
\bigm| \bigm| + \bigm| \bigm| f \prime (b)
\bigm| \bigm| \bigr) (1.5)
for \alpha > 0.
Remark 1.1. (1) The assumptions f : [a, b] \rightarrow \BbbR is positive with 0 \leq a < b in Theorem E can
be weakened as f : [a, b] \rightarrow \BbbR with a < b.
(2) In Theorem D, let q = 1. Then Theorem D reduces to Theorems B and C.
(3) In Theorems F and G, let \alpha = 1. Then Theorems F and G reduce to Theorem B and C,
respectively.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
410 K.-L. TSENG, K.-C. HSU
(4) In Theorem H, let \alpha = 1. Then the inequality (1.3) is connected with the second inequality
in the inequality (1.2).
(5) In Theorem I, let \alpha = 1. Then the inequality (1.4) is a Bullen-type inequality.
(6) In Theorem J, let \alpha = 1. Then the inequality (1.5) is a Simpson-type inequality.
In this paper, we establish some new Hermite – Hadamard-type inequalities for fractional integrals
which generalize Theorems D and G – J. Some applications for the Beta-function are given.
2. Main results.
Theorem 2.1. Under the assumptions of Theorem D, then we have the following Hermite –
Hadamard-type inequality for fractional integrals:\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(\alpha + 1)
\biggl(
\alpha - 1 +
1
2\alpha - 1
\biggr) \biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
(2.1)
for \alpha > 0.
Proof. In [9], let
h1(x) =
\left\{
(b - x)\alpha - (x - a)\alpha - (b - a)\alpha , x \in
\biggl[
a,
a+ b
2
\biggr)
,
(b - x)\alpha - (x - a)\alpha + (b - a)\alpha , x \in
\biggl[
a+ b
2
, b
\biggr]
.
Then the following identities hold:
1
2(b - a)\alpha
b\int
a
h1(x)f
\prime (x) dx =
=
\alpha
2(b - a)\alpha
b\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
f(x) dx - f
\biggl(
a+ b
2
\biggr)
=
=
\alpha \Gamma (\alpha )
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - (a)] - f
\biggl(
a+ b
2
\biggr)
=
=
\Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - (a)] - f
\biggl(
a+ b
2
\biggr)
. (2.2)
Using simple computation, we have the following identities:
x =
b - x
b - a
a+
x - a
b - a
b, x \in [a, b], (2.3)
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 411
+
b\int
a+b
2
[(b - x)\alpha - (x - a)\alpha + (b - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ] dx := M1, (2.4)
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
b\int
a+b
2
[(b - x)\alpha - (x - a)\alpha + (b - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ] dx := M2, (2.5)
b\int
a
| h1(x)| dx = 2
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ] dx. (2.6)
Now, using power mean inequality, the identities (2.3) – (2.6) and the convexity of | f \prime | q, we obtain
the inequality
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
b\int
a
h1(x)f
\prime (x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
b\int
a
| h1(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx \leq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
412 K.-L. TSENG, K.-C. HSU
\leq
\left[ b\int
a
| h1(x)| dx
\right]
q - 1
q
\left[ b\int
a
| h1(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx
\right] 1/q
=
=
\left[ b\int
a
| h1(x)| dx
\right]
q - 1
q
\left[
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx +
+
b\int
a+b
2
[(b - x)\alpha - (x - a)\alpha + (b - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx
\right]
1/q
\leq
\leq
\left[ b\int
a
| h1(x)| dx
\right]
q - 1
q
(M1 +M2)
1/q =
= 2
a+b
2\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ] dx
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
=
=
(b - a)\alpha +1
\alpha + 1
\biggl(
\alpha - 1 +
1
2\alpha - 1
\biggr) \biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
. (2.7)
The inequality (2.1) follows from the identity (2.2) and the inequality (2.7).
Theorem 2.1 is proved.
Remark 2.1. In Theorem 2.1, let q = 1. Then Theorem 2.1 reduces to Theorem G.
Theorem 2.2. Under the assumptions of Theorem D, then we have the following inequality for
fractional integrals with
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr)
2
:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] -
1
2
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl(
1
4
+
3\alpha +1 - 2\alpha +1 + 1
2 \cdot 4\alpha (\alpha + 1)
- 1
\alpha + 1
\biggr)
(b - a)
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
(2.8)
for \alpha > 0.
Proof. In [9], let
h2(x) =
\left\{
(b - x)\alpha - (x - a)\alpha - (b - a)\alpha , x \in
\biggl[
a,
3a+ b
4
\biggr)
,
(b - x)\alpha - (x - a)\alpha , x \in
\biggl[
3a+ b
4
,
a+ 3b
4
\biggr)
,
(b - x)\alpha - (x - a)\alpha + (b - a)\alpha , x \in
\biggl[
a+ 3b
4
, b
\biggr]
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 413
Then the following identities hold:
1
2(b - a)\alpha
b\int
a
h2(x)f
\prime (x) dx =
=
\alpha
2(b - a)\alpha
b\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
f(x) dx -
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr)
2
=
=
\alpha \Gamma (\alpha )
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - (a)] -
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr)
2
=
=
\Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - (a)] -
1
2
\biggl[
f
\biggl(
3a+ b
4
\biggr)
+ f
\biggl(
a+ 3b
4
\biggr) \biggr]
. (2.9)
Using simple computation, we get the identities
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
b\int
a+3b
4
[(b - x)\alpha - (x - a)\alpha + (b - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q 3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ] dx := N1, (2.10)
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
b\int
a+3b
4
[(b - x)\alpha - (x - a)\alpha + (b - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
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414 K.-L. TSENG, K.-C. HSU
+
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q 3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ] dx := N2, (2.11)
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
a+3b
4\int
a+b
2
[(x - a)\alpha - (b - x)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ] dx := N3, (2.12)
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
a+3b
4\int
a+b
2
[(x - a)\alpha - (b - x)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ]
b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
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NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 415
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ] dx := N4, (2.13)
b\int
a
| h2(x)| dx = 2
\left[
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ] dx+
+
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ] dx
\right] . (2.14)
Now, using power mean inequality, the identities (2.3), (2.10) – (2.14) and the convexity of | f \prime | q,
we have the inequality
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
b\int
a
h2(x)f
\prime (x) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
b\int
a
| h2(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx \leq
\leq
\left[ b\int
a
| h2(x)| dx
\right]
q - 1
q
\left[ b\int
a
| h2(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx
\right] 1/q
=
=
\left[ b\int
a
| h2(x)| dx
\right]
q - 1
q
\left[
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx+
+
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx+
+
a+3b
4\int
a+b
2
[(x - a)\alpha - (b - x)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx+
+
b\int
a+3b
4
[(b - x)\alpha - (x - a)\alpha + (b - a)\alpha ]
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx
\right]
1/q
\leq
\leq
\left[ b\int
a
| h2(x)| dx
\right]
q - 1
q
(N1 +N2 +N3 +N4) =
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416 K.-L. TSENG, K.-C. HSU
= 2
\left[
3a+b
4\int
a
[(b - a)\alpha - (b - x)\alpha + (x - a)\alpha ] dx+
+
a+b
2\int
3a+b
4
[(b - x)\alpha - (x - a)\alpha ] dx
\right] \biggl[ | f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
=
=
\biggl(
1
2
+
3\alpha +1 - 2\alpha +1 + 1
4\alpha (\alpha + 1)
- 2
\alpha + 1
\biggr)
(b - a)\alpha +1
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
. (2.15)
The inequality (2.8) follows from the identity (2.9) and the inequality (2.15).
Theorem 2.2 is proved.
Remark 2.2. (1) In Theorem 2.2, let q = 1. Then Theorem 2.2 reduces to Theorem H.
(2) In Theorems 2.1 and 2.2, let \alpha = 1. Then Theorems 2.1 and 2.2 reduce to Theorem D.
Theorem 2.3. Under the assumptions of Theorem D, then we have the following Bullen-type
inequality for fractional integrals:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] -
-
\biggl[
3\alpha - 1
4\alpha
f
\biggl(
a+ b
2
\biggr)
+
4\alpha - 3\alpha + 1
4\alpha
f(a) + f(b)
2
\biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
\alpha + 1
\biggl(
2\alpha + 1
2\alpha
- 3\alpha +1 + 1
2 \cdot 4\alpha
\biggr)
(b - a)
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
(2.16)
for \alpha > 0.
Proof. Let
h3(x) =
\left\{
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha , x \in
\biggl[
a,
a+ b
2
\biggr)
,
(b - x)\alpha - (x - a)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha , x \in
\biggl[
a+ b
2
, b
\biggr]
.
Then the following identities hold:
1
2(b - a)\alpha
b\int
a
h3(x)f
\prime (x) dx =
=
\alpha
2(b - a)\alpha
b\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
f(x) dx -
-
\biggl[
3\alpha - 1
4\alpha
f
\biggl(
a+ b
2
\biggr)
+
4\alpha - 3\alpha + 1
4\alpha
f(a) + f(b)
2
\biggr]
=
=
\alpha \Gamma (\alpha )
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - (a)] -
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NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 417
-
\biggl[
3\alpha - 1
4\alpha
f
\biggl(
a+ b
2
\biggr)
+
4\alpha - 3\alpha + 1
4\alpha
f(a) + f(b)
2
\biggr]
=
=
\Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - (a)] -
-
\biggl[
3\alpha - 1
4\alpha
f
\biggl(
a+ b
2
\biggr)
+
4\alpha - 3\alpha + 1
4\alpha
f(a) + f(b)
2
\biggr]
. (2.17)
Using simple computation, we have the following identities:
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
b\int
a+3b
4
\biggl[
(x - a)\alpha - (b - x)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q 3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
dx := P1, (2.18)
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
b\int
a+3b
4
\biggl[
(x - a)\alpha - (b - x)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q 3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
dx := P2, (2.19)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
418 K.-L. TSENG, K.-C. HSU
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
a+3b
4\int
a+b
2
\biggl[
(b - x)\alpha - (x - a)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
dx := P3, (2.20)
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
a+3b
4\int
a+b
2
\biggl[
(b - x)\alpha - (x - a)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
dx := P4, (2.21)
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NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 419
b\int
a
| h3(x)| dx = 2
\left[
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
dx+
+
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
dx
\right] . (2.22)
Now, using power mean inequality, the identities (2.3), (2.18) – (2.22) and the convexity of | f \prime | q,
we obtain the inequality
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
b\int
a
h3(x)f
\prime (x)dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
b\int
a
| h3(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx \leq
\leq
\left[ b\int
a
| h3(x)| dx
\right]
q - 1
q
\left[ b\int
a
| h3(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx
\right] 1/q
=
=
\left[ b\int
a
| h3(x)| dx
\right]
q - 1
q
\left[
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
\biggr]
(b - a)\alpha
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx +
+
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx+
+
a+3b
4\int
a+b
2
\biggl[
(b - x)\alpha - (x - a)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx+
+
b\int
a+3b
4
\biggl[
(x - a)\alpha - (b - x)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx
\right]
1/q
\leq
\leq
\left[ b\int
a
| h3(x)| dx
\right]
q - 1
q
(P1 + P2 + P3 + P4) =
= 2
\left[
3a+b
4\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 3\alpha - 1
4\alpha
(b - a)\alpha
\biggr]
dx +
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420 K.-L. TSENG, K.-C. HSU
+
a+b
2\int
3a+b
4
\biggl[
(x - a)\alpha - (b - x)\alpha +
3\alpha - 1
4\alpha
(b - a)\alpha
\biggr] \right] dx \biggl[ | f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
=
=
1
\alpha + 1
\biggl(
2\alpha + 1
2\alpha - 1
- 3\alpha +1 + 1
4\alpha
\biggr)
(b - a)\alpha +1
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
. (2.23)
The inequality (2.16) follows from the identity (2.17) and the inequality (2.23).
Theorem 2.3 is proved.
Remark 2.3. In Theorem 2.3, let q = 1. Then Theorem 2.3 reduces to Theorem I.
Theorem 2.4. Under the assumptions of Theorem D, then we have the following Simpson-type
inequality for fractional integrals:\bigm| \bigm| \bigm| \bigm| \Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - f(a)] -
-
\biggl[
5\alpha - 1
6\alpha
f
\biggl(
a+ b
2
\biggr)
+
6\alpha - 5\alpha + 1
6\alpha
f(a) + f(b)
2
\biggr] \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl[
1
\alpha + 1
\biggl(
2\alpha + 1
2\alpha
- 5\alpha +1 + 1
3 \cdot 6\alpha
\biggr)
+
\biggl(
5\alpha - 1
6\alpha +1
\biggr) \biggr]
(b - a)
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
(2.24)
for \alpha > 0.
Proof. In [9], let
h4(x) =
\left\{
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha , x \in
\biggl[
a,
a+ b
2
\biggr)
,
(b - x)\alpha - (x - a)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha , x \in
\biggl[
a+ b
2
, b
\biggr]
.
Then, the following identities hold:
1
2(b - a)\alpha
b\int
a
h4(x)f
\prime (x) dx =
=
\alpha
2(b - a)\alpha
b\int
a
\bigl[
(x - a)\alpha - 1 + (b - x)\alpha - 1
\bigr]
f(x) dx -
-
\biggl[
5\alpha - 1
6\alpha
f
\biggl(
a+ b
2
\biggr)
+
6\alpha - 5\alpha + 1
6\alpha
f(a) + f(b)
2
\biggr]
=
=
\alpha \Gamma (\alpha )
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - (a)] -
-
\biggl[
5\alpha - 1
6\alpha
f
\biggl(
a+ b
2
\biggr)
+
6\alpha - 5\alpha + 1
6\alpha
f(a) + f(b)
2
\biggr]
=
=
\Gamma (\alpha + 1)
2(b - a)\alpha
[J\alpha
a+f(b) + J\alpha
b - (a)] -
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NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 421
-
\biggl[
5\alpha - 1
6\alpha
f
\biggl(
a+ b
2
\biggr)
+
6\alpha - 5\alpha + 1
6\alpha
f(a) + f(b)
2
\biggr]
. (2.25)
Using simple computation, we have the following identities:
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
b\int
a+5b
6
\biggl[
(x - a)\alpha - (b - x)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q 5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
dx := Q1, (2.26)
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
b\int
a+5b
6
\biggl[
(x - a)\alpha - (b - x)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q 5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
dx := Q2, (2.27)
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
422 K.-L. TSENG, K.-C. HSU
+
a+5b
6\int
a+b
2
\biggl[
(b - x)\alpha - (x - a)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx+
+
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (a)
\bigm| \bigm| q a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
dx := Q3, (2.28)
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
a+5b
6\int
a+b
2
\biggl[
(b - x)\alpha - (x - a)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
x - a
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx+
+
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
b - x
b - a
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q dx =
=
\bigm| \bigm| f \prime (b)
\bigm| \bigm| q a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
dx := Q4, (2.29)
b\int
a
| h4(x)| dx = 2
\left[
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
dx+
+
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
dx
\right] . (2.30)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 423
Now, using power mean inequality, the identities (2.3), (2.26) – (2.30) and the convexity of | f \prime | q,
we get the inequality \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
b\int
a
h4(x)f
\prime (x)dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
b\int
a
| h4(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| dx \leq
\leq
\left[ b\int
a
| h4(x)| dx
\right]
q - 1
q
\left[ b\int
a
| h4(x)|
\bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx
\right] 1/q
=
=
\left[ b\int
a
| h4(x)| dx
\right]
q - 1
q
\left[
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx +
+
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx+
+
a+5b
6\int
a+b
2
\biggl[
(b - x)\alpha - (x - a)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx+
+
b\int
a+5b
6
\biggl[
(x - a)\alpha - (b - x)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr] \bigm| \bigm| f \prime (x)
\bigm| \bigm| q dx
\right] \leq
\leq
\left[ b\int
a
| h4(x)| dx
\right]
q - 1
q
(Q1 +Q2 +Q3 +Q4) =
= 2
\left[
5a+b
6\int
a
\biggl[
(b - x)\alpha - (x - a)\alpha - 5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
dx+
+
a+b
2\int
5a+b
6
\biggl[
(x - a)\alpha - (b - x)\alpha +
5\alpha - 1
6\alpha
(b - a)\alpha
\biggr]
dx
\right] \biggl[ | f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
=
=
\biggl[
1
\alpha + 1
\biggl(
2\alpha + 1
2\alpha - 1
- 5\alpha +1 + 1
9 \cdot 6\alpha - 1
\biggr)
+
\biggl(
5\alpha - 1
3 \cdot 6\alpha
\biggr) \biggr]
(b - a)\alpha +1
\biggl[
| f \prime (a)| q + | f \prime (b)| q
2
\biggr] 1/q
. (2.31)
The inequality (2.24) follows from the identity (2.25) and the inequality (2.31).
Theorem 2.4 is proved.
Remark 2.4. In Theorem 2.4, let q = 1. Then Theorem 2.4 reduces to Theorem J.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
424 K.-L. TSENG, K.-C. HSU
3. Applications for the Beta-functions. Throughout this section, let \alpha > 0, \rho \geq 1, q \geq 1,
a = 0, b = 1, \Gamma (\alpha ) be the Gamma-function and f(x) = x\rho - 1 (x \in [0, 1]) . Then | f \prime | is convex on
[0, 1].
Let us recall the Beta-function
B (p, r) =
1\int
0
xp - 1 (1 - x)r - 1 dx (p, r > 0) .
Remark 3.1. Using Theorems 2.1 – 2.4, we get
\Gamma (\alpha + 1)
2(b - a)\alpha
J\alpha
a+f(b) =
\alpha
2
1\int
0
(1 - x)\alpha - 1 x\rho - 1dx =
\alpha
2
B (\rho , \alpha )
and
\Gamma (\alpha + 1)
2(b - a)\alpha
J\alpha
b - f(a) =
\alpha
2
1\int
0
x\alpha +\rho - 2dx =
\alpha
2 (\alpha + \rho - 1)
.
Using Theorems 2.1 – 2.4 and Remark 3.1, we have the following propositions:
Proposition 3.1. In Theorem 2.1, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \alpha 2B (\rho , \alpha ) +
\alpha
2 (\alpha + \rho - 1)
- 1
2\rho - 1
\bigm| \bigm| \bigm| \bigm| \leq \biggl(
1
2
- 2\alpha - 1
2\alpha (\alpha + 1)
\biggr)
\rho - 1
21/q
.
Proposition 3.2. In Theorem 2.2, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \alpha 2B (\rho , \alpha ) +
\alpha
2 (\alpha + \rho - 1)
- 3\rho - 1 + 1
2 \cdot 4\rho - 1
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl[
3\alpha +1 + 1
2 \cdot 4\alpha (\alpha + 1)
+
1
4
- 2\alpha + 1
2\alpha (\alpha + 1)
\biggr]
\rho - 1
21/q
.
Proposition 3.3. In Theorem 2.3, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \alpha 2B (\rho , \alpha ) +
\alpha
2 (\alpha + \rho - 1)
-
\biggl(
3\alpha - 1
2\rho - 14\alpha
+
4\alpha - 3\alpha + 1
2 \cdot 4\alpha
\biggr) \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
\alpha + 1
\biggl(
2\alpha + 1
2\alpha
- 3\alpha +1 + 1
2 \cdot 4\alpha
\biggr)
\rho - 1
21/q
.
Proposition 3.4. In Theorem 2.4, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \alpha 2B (\rho , \alpha ) +
\alpha
2 (\alpha + \rho - 1)
-
\biggl(
5\alpha - 1
2\rho - 16\alpha
+
6\alpha - 5\alpha + 1
2 \cdot 6\alpha
\biggr) \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl[
1
\alpha + 1
\biggl(
2\alpha + 1
2\alpha
- 5\alpha +1 + 1
3 \cdot 6\alpha
\biggr)
+
\biggl(
5\alpha - 1
6\alpha +1
\biggr) \biggr]
\rho - 1
21/q
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
NEW FRACTIONAL INTEGRAL INEQUALITIES FOR DIFFERENTIABLE CONVEX FUNCTIONS . . . 425
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Received 18.11.15
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 3
|
| id | umjimathkievua-article-1705 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:11:00Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/39/232651fd32fe5ee207b0827b2531fa39.pdf |
| spelling | umjimathkievua-article-17052019-12-05T09:24:16Z New fractional integral inequalities for differentiable convex functions and their applications Новi дробово-iнтегральнi нерiвностi для диференцiйовних опуклих функцiй та їх застосування Hsu, K.-C. Tseng, K.-L. Хсу, К.-С. Цзен, К.-Л. We establish some new fractional integral inequalities for differentiable convex functions and give several applications for the Beta-function. Встановлено деякi новi дробово-iнтегральнi нерiвностi для диференцiйовних опуклих функцiй i наведено кiлька застосувань для бета-функцiї. Institute of Mathematics, NAS of Ukraine 2017-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1705 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 3 (2017); 407-425 Український математичний журнал; Том 69 № 3 (2017); 407-425 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1705/687 Copyright (c) 2017 Hsu K.-C.; Tseng K.-L. |
| spellingShingle | Hsu, K.-C. Tseng, K.-L. Хсу, К.-С. Цзен, К.-Л. New fractional integral inequalities for differentiable convex functions and their applications |
| title | New fractional integral inequalities for differentiable convex functions and
their applications |
| title_alt | Новi дробово-iнтегральнi нерiвностi для диференцiйовних опуклих функцiй та їх застосування |
| title_full | New fractional integral inequalities for differentiable convex functions and
their applications |
| title_fullStr | New fractional integral inequalities for differentiable convex functions and
their applications |
| title_full_unstemmed | New fractional integral inequalities for differentiable convex functions and
their applications |
| title_short | New fractional integral inequalities for differentiable convex functions and
their applications |
| title_sort | new fractional integral inequalities for differentiable convex functions and
their applications |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1705 |
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