Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set
UDC 517.5 The authors derive a fractional integral equality concerning twice differentiable mappings defined on $m$-invex set. By using this identity, the authors obtain new estimates on generalization of trapezium-like inequalities for mappings whose second order derivatives are generalized relativ...
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Institute of Mathematics, NAS of Ukraine
2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512240360226816 |
|---|---|
| author | Du, T. S. Luo , C. Y. Huang , Z. Z. Kashuri , A. Du, T. S. Luo , C. Y. Huang , Z. Z. Kashuri , A. A. |
| author_facet | Du, T. S. Luo , C. Y. Huang , Z. Z. Kashuri , A. Du, T. S. Luo , C. Y. Huang , Z. Z. Kashuri , A. A. |
| author_sort | Du, T. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:49:28Z |
| description | UDC 517.5
The authors derive a fractional integral equality concerning twice differentiable mappings defined on $m$-invex set. By using this identity, the authors obtain new estimates on generalization of trapezium-like inequalities for mappings whose second order derivatives are generalized relative semi-$(m, h_1, h_2)$-preinvex via fractional integrals. We also discuss some new special cases which can be deduced from our main results. |
| doi_str_mv | 10.37863/umzh.v72i12.6036 |
| first_indexed | 2026-03-24T03:25:38Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i12.6036
UDC 517.5
T. S. Du (College Sci., China Three Gorges Univ. and Three Gorges Math. Res. Center, Yichang, China),
C. Y. Luo, Z. Z. Huang (College Sci., China Three Gorges Univ., Yichang, China),
A. Kashuri (Univ. “Ismail Qemali”, Vlora, Albania)
FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES INVOLVING
GENERALIZED RELATIVE SEMI-(\bfitm ,\bfith \bfone , \bfith \bftwo )-PREINVEX MAPPINGS
ON AN \bfitm -INVEX SET
ДРОБОВI НЕРIВНОСТI ТИПУ ТРАПЕЦIЇ З УЗАГАЛЬНЕНИМИ
ВIДНОСНО НАПIВ-(\bfitm ,\bfith \bfone , \bfith \bftwo )-ПРЕIНВЕКСНИМИ ВIДОБРАЖЕННЯМИ
НА \bfitm -IНВЕКСНIЙ МНОЖИНI
The authors derive a fractional integral equality concerning twice differentiable mappings defined on m-invex set. By using
this identity, the authors obtain new estimates on generalization of trapezium-like inequalities for mappings whose second
order derivatives are generalized relative semi-(m,h1, h2)-preinvex via fractional integrals. We also discuss some new
special cases which can be deduced from our main results.
Встановлено дробову iнтегральну рiвнiсть для двiчi диференцiйовних вiдображень на m-iнвекснiй множинi. За
допомогою цiєї рiвностi отримано новi оцiнки для узагальнених нерiвностей типу трапецiї для вiдображень, у яких
похiднi другого порядку є узагальненими вiдносно напiв-(m,h1, h2)-преiнвексними через дробовi iнтеграли. Також
обговорено деякi новi спецiальнi випадки, що випливають з отриманих результатiв.
1. Introduction. In [19], Sarikaya et al. established the following interesting Hadamard-type
inequalities by using Riemann – Liouville fractional integrals.
Theorem 1.1. Let f : [u, v] \rightarrow \BbbR be a positive mapping along with 0 \leq u < v and let f belongs
to L1[u, v]. Suppose that f is a convex function on [u, v], then the following double inequalities for
fractional integrals hold:
f
\biggl(
u+ v
2
\biggr)
\leq \Gamma (\alpha + 1)
2(v - u)\alpha
[J\alpha
u+f(v) + J\alpha
v - f(u)] \leq
f(u) + f(v)
2
, (1.1)
where the symbols J\alpha
u+f and J\alpha
v - f denote, respectively, the left- and right-sided Riemann – Liouville
fractional integrals of order \alpha \in \BbbR + defined by
J\alpha
u+f(x) =
1
\Gamma (\alpha )
x\int
u
(x - t)\alpha - 1f(t)dt, u < x,
and
J\alpha
v - f(x) =
1
\Gamma (\alpha )
v\int
x
(t - x)\alpha - 1f(t)dt, x < v.
Here, \Gamma (\alpha ) is the gamma function and its definition is \Gamma (\alpha ) =
\int \infty
0
e - \mu \mu \alpha - 1d\mu .
c\bigcirc T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12 1633
1634 T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI
For \alpha = 1, the inequality (1.1) reduces to the successive Hadamard-type inequality
f
\biggl(
u+ v
2
\biggr)
\leq 1
v - u
v\int
u
f(x)dx \leq f(u) + f(v)
2
, (1.2)
where f : I \subseteq \BbbR \rightarrow \BbbR is a convex mapping on the interval I of real numbers and u, v \in I with
u < v. This inequality (1.2) is also named as trapezium inequality.
In recent years, many researchers have studied error bounds with respect to the inequality (1.2);
for refinements, counterparts, generalization please refer to [1, 4, 6, 11, 12, 16, 21, 22, 24] and
references cited therein.
More integral inequalities via fractional integrals may be seen in [2, 7 – 10, 13, 14, 18, 20].
Our goal is to establish, employing the Riemann – Liouville fractional calculus, some new left-
sided Hadamard-type integral inequalities. We deal with mappings which have absolute values of
the second derivatives which are generalized relative semi-(m,h1, h2)-preinvex. These inequalities
obtained in this paper can be viewed as generalization of the results of [14] and [15].
To end this section, we evoke some special functions and definitions as follows:
(1) the beta function
\beta (x, y) =
1\int
0
tx - 1(1 - t)y - 1dt =
\Gamma (x)\Gamma (y)
\Gamma (x+ y)
, x, y > 0,
(2) the hypergeometric function
2F1(x, y; c; z) =
1
\beta (y, c - y)
1\int
0
ty - 1(1 - t)c - y - 1(1 - zt) - xdt
for | z| < 1, c > y > 0.
Definition 1.1 [5]. A set K \subseteq \BbbR n is said to be m-invex with respect to the mapping \eta : K \times
\times K \times (0, 1] \rightarrow \BbbR n for some fixed m \in (0, 1], if mx + t\eta (y, x,m) \in K holds for each x, y \in K
and any t \in [0, 1].
Definition 1.2 [17]. Let K \subseteq \BbbR be an open m-invex set with respect to \eta : K\times K\times (0, 1] \rightarrow \BbbR
and let h1, h2 : [0, 1] \rightarrow \BbbR 0. A function f : K \rightarrow \BbbR is said to be generalized (m,h1, h2)-preinvex if
the inequality
f (mx+ t\eta (y, x,m)) \leq mh1(t)f(x) + h2(t)f(y)
is valid for all x, y \in K and t \in [0, 1].
Definition 1.3 [23]. A set M\varphi \subseteq \BbbR n is named a relative convex (\varphi -convex) set if and only if
there exists a function \varphi : \BbbR n \rightarrow \BbbR n such that
t\varphi (x) + (1 - t)\varphi (y) \in M\varphi \forall x, y \in \BbbR n : \varphi (x), \varphi (y) \in M\varphi , t \in [0, 1].
Definition 1.4 [23]. A function f is named a relative convex (\varphi -convex) function on a relative
convex (\varphi -convex) set M\varphi if and only if there exists a function \varphi : \BbbR n \rightarrow \BbbR n such that,
f(t\varphi (x) + (1 - t)\varphi (y)) \leq tf(\varphi (x)) + (1 - t)f(\varphi (y))
\forall x, y \in \BbbR n : \varphi (x), \varphi (y) \in M\varphi , t \in [0, 1].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES . . . 1635
Definition 1.5 [3]. A function f is said to be a relative semiconvex on a relative convex set M\varphi
if and only if there exists a function \varphi : \BbbR n \rightarrow \BbbR n such that
f(t\varphi (x) + (1 - t)\varphi (y)) \leq tf(x) + (1 - t)f(y) \forall x, y \in M\varphi , t \in [0, 1].
2. Auxiliary results. We begin with the following new definition.
Definition 2.1. Let K \subseteq \BbbR be an open nonempty m-invex set with respect to the mapping \eta :
K \times K \times (0, 1] \rightarrow \BbbR and \varphi : I \rightarrow K is a continuous function. A function f : K \rightarrow \BbbR , h1, h2 :
[0, 1] \rightarrow \BbbR 0 is said to be generalized relative semi-(m,h1, h2)-preinvex functions if
f (m\varphi (x) + t\eta (\varphi (y), \varphi (x),m)) \leq mh1(t)f(x) + h2(t)f(y) (2.1)
holds for all x, y \in I and t \in [0, 1] with some fixed m \in (0, 1].
Remark 2.1. In Definition 2.1, let us discuss some special cases as follows:
(I) if we take h1(t) = (1 - t)s and h2(t) = ts for s \in (0, 1], then we get generalized relative
semi-(m, s)-Breckner-preinvex functions,
(II) if we take h1(t) = h2(t) = 1, then we get generalized relative semi-(m,P )-preinvex
functions,
(III) if we take h1(t) = (1 - t) - s and h2(t) = t - s for s \in (0, 1], then we get generalized relative
semi-(m, s)-Godunova – Levin – Dragomir-preinvex functions,
(IV) if we take h1(t) = h(1 - t) and h2(t) = h(t), then we get generalized relative semi-(m,h)-
preinvex functions,
(V) if we take h1(t) = h2(t) = t(1 - t), then we get generalized relative semi-(m, tgs)-preinvex
functions,
(VI) if we take h1(t) =
\surd
1 - t
2
\surd
t
and h2(t) =
\surd
t
2
\surd
1 - t
, then we get generalized relative semi-
m-MT -preinvex functions.
It is worth to mention here that to the best of our knowledge all the special cases discussed above
are new in the literature.
Throughout of this paper, let \varphi : I \subseteq \BbbR \rightarrow K be a continuous function with a, b \in I, \varphi (a) <
< \varphi (b) and let K \subseteq \BbbR be an open nonempty m-invex subset with respect to \eta : K \times K \times (0, 1] \rightarrow
\rightarrow \BbbR for some fixed m \in (0, 1] with 0 < \eta (\varphi (b), \varphi (a),m). Suppose that f : K \rightarrow \BbbR be a twice
differentiable function on the interior K\circ of K and f \prime \prime \in L1 [m\varphi (a),m\varphi (a) + \eta (\varphi (b), \varphi (a),m)] .
Before stating the results we use following notations:
G(\alpha ;n,m,\varphi (a), \varphi (b))(f) =
(n+ 1)\alpha \Gamma (\alpha + 2)
\eta \alpha (\varphi (b), \varphi (a),m)
\biggl[
J\alpha
(m\varphi (a)+ 1
n+1
\eta (\varphi (b),\varphi (a),m)) -
f(m\varphi (a))+
+J\alpha
(m\varphi (a)+ n
n+1
\eta (\varphi (b),\varphi (a),m))+f (m\varphi (a) + \eta (\varphi (b), \varphi (a),m))
\biggr]
-
- \eta (\varphi (b), \varphi (a),m)
n+ 1
\biggl[
f \prime
\biggl(
m\varphi (a) +
n
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
-
- f \prime
\biggl(
m\varphi (a) +
1
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \biggr]
-
- (\alpha + 1)
\biggl[
f
\biggl(
m\varphi (a) +
n
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
+ f
\biggl(
m\varphi (a) +
1
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \biggr]
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1636 T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI
Lemma 2.1. We have the following identity for fractional integrals along with
x \in [m\varphi (a),m\varphi (a) + \eta (\varphi (b), \varphi (a),m)] , \alpha > 0 and n \in \BbbN + :
G(\alpha ;n,m,\varphi (a), \varphi (b))(f) =
=
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
1\int
0
(1 - t)\alpha +1
\biggl[
f \prime \prime
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
+
+f \prime \prime
\biggl(
m\varphi (a) +
n+ t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \biggr]
dt. (2.2)
Proof. Let
I\ast =
1\int
0
(1 - t)\alpha +1
\biggl[
f \prime \prime
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
+
+f \prime \prime
\biggl(
m\varphi (a) +
n+ t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \biggr]
dt =
=
1\int
0
(1 - t)\alpha +1f \prime \prime
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
dt+
+
1\int
0
(1 - t)\alpha +1f \prime \prime
\biggl(
m\varphi (a) +
n+ t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
dt := I1 + I2. (2.3)
Integrating I1 on [0, 1] yields
I1 =
n+ 1
\eta (\varphi (b), \varphi (a),m)
f \prime
\biggl(
m\varphi (a) +
1
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
-
- (n+ 1)2(\alpha + 1)
\eta 2(\varphi (b), \varphi (a),m)
f
\biggl(
m\varphi (a) +
1
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
+
+
(n+ 1)2\alpha (\alpha + 1)
\eta 2(\varphi (b), \varphi (a),m)
1\int
0
(1 - t)\alpha - 1f
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
dt =
=
n+ 1
\eta (\varphi (b), \varphi (a),m)
f \prime
\biggl(
m\varphi (a) +
1
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
-
- (n+ 1)2(\alpha + 1)
\eta 2(\varphi (b), \varphi (a),m)
f
\biggl(
m\varphi (a) +
1
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
+
+
(n+ 1)\alpha +2\Gamma (\alpha + 2)
\eta (\varphi (b), \varphi (a),m)\alpha +2
J\alpha
(m\varphi (a)+ 1
n+1
\eta (\varphi (b),\varphi (a),m)) -
f(m\varphi (a)). (2.4)
Analogously, integrating I2 on [0, 1], we have
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES . . . 1637
I2 = - n+ 1
\eta (\varphi (b), \varphi (a),m)
f \prime
\biggl(
m\varphi (a) +
n
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
-
- (n+ 1)2(\alpha + 1)
\eta 2(\varphi (b), \varphi (a),m)
f
\biggl(
m\varphi (a) +
n
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr)
+
+
(n+ 1)\alpha +2\Gamma (\alpha + 2)
\eta (\varphi (b), \varphi (a),m)\alpha +2
J\alpha
(m\varphi (a)+ n
n+1
\eta (\varphi (b),\varphi (a),m))+f(m\varphi (a) + \eta (\varphi (b), \varphi (a),m)). (2.5)
Applying (2.4) and (2.5) to (2.3), then multiplying both sides by
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
ends the proof.
Remark 2.2. If \eta (\varphi (b), \varphi (a),m) = \varphi (b) - m\varphi (a) with m = 1 and \varphi is an identity mapping,
then Lemma 2.1 reduces to Lemma 1.3 in [14]. Further, if we put n = 1, then we obtain Lemma 1
in [15].
3. Main results. Using Lemma 2.1, we now state the following theorem.
Theorem 3.1. If | f \prime \prime | q for q \geq 1 is generalized relative semi-(m,h1, h2)-preinvex, then the
following inequality for Riemann – Liouville fractional integrals along with h1, h2 : [0, 1] \rightarrow \BbbR 0,
\alpha > 0 and n \in \BbbN + exists:\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
1
\alpha + 2
\biggr) 1 - 1
q
\times
\times
\left\{
\left[ 1\int
0
(1 - t)\alpha +1
\biggl(
mh1
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
\left[ 1\int
0
(1 - t)\alpha +1
\biggl(
mh1
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
\right\} . (3.1)
Proof. Using given hypothesis, Lemma 2.1 and the power mean inequality, we have\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq
\leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
1\int
0
(1 - t)\alpha +1
\bigm| \bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| \bigm| dt+
+
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
1\int
0
(1 - t)\alpha +1
\bigm| \bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
n+ t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| \bigm| dt \leq
\leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\left( 1\int
0
(1 - t)\alpha +1dt
\right) 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\alpha +1
\bigm| \bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| \bigm| qdt
\right]
1
q
+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1638 T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI
+
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\left( 1\int
0
(1 - t)\alpha +1dt
\right) 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\alpha +1
\bigm| \bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
n+ t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| \bigm| qdt
\right]
1
q
\leq
\leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
1
\alpha + 2
\biggr) 1 - 1
q
\times
\times
\left\{
\left[ 1\int
0
(1 - t)\alpha +1
\biggl(
mh1
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
\left[ 1\int
0
(1 - t)\alpha +1
\biggl(
mh1
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
\right\} ,
which completes the proof.
Remark 3.1. In Theorem 3.1, if we take h1(t) = (1 - t)s, h2(t) = ts and \varphi is an identity
mapping along with \eta (\varphi (b), \varphi (a),m) = \varphi (b) - m\varphi (a), m = 1, then we have:
(a) for q = 1, we get Theorem 2.1 in [14], specially, for n = 1, we obtain Theorem 2 in [15],
(b) for n = 1, we have Theorem 4 in [15],
(c) for n > 1, we get Theorem 2.4 in [14].
Corollary 3.1. In Theorem 3.1, if we put h1(t) = (1 - t) - s and h2(t) = t - s, then we have:
(1) for n > 1, the following inequality for generalized relative semi-(m, s)-Godunova – Levin –
Dragomir-preinvex functions holds:\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)
2 - s
q
\biggl(
1
\alpha + 2
\biggr) 1 - 1
q
\times
\times
\left\{
\left[ mn - s
2F1
\biggl[
s, 1;\alpha + 3; - 1
n
\biggr]
\alpha + 2
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1
\alpha - s+ 2
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
+
\left[ m
\alpha - s+ 2
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + n - s
2F1
\biggl[
s, 1;\alpha + 3; - 1
n
\biggr]
\alpha + 2
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} ,
(2) for n = 1, the following inequality for generalized relative semi-(m, s)-Godunova – Levin –
Dragomir-preinvex functions exists:
1
2(\alpha + 1)
\bigm| \bigm| \bigm| G(\alpha ; 1,m, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| =
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FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES . . . 1639
=
\bigm| \bigm| \bigm| \bigm| \bigm| 2\alpha - 1\Gamma (\alpha + 1)
\eta \alpha (\varphi (b), \varphi (a),m)
\Bigl[
J\alpha
(m\varphi (a)+ 1
2
\eta (\varphi (b),\varphi (a),m)) -
f(m\varphi (a))+
+J\alpha
(m\varphi (a)+ 1
2
\eta (\varphi (b),\varphi (a),m))+
f (m\varphi (a) + \eta (\varphi (b), \varphi (a),m))
\Bigr]
-
\biggl[
f
\biggl(
m\varphi (a) +
1
2
\eta (\varphi (b), \varphi (a),m)
\biggr) \biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \eta 2(\varphi (b), \varphi (a),m)
2
3 - s
q (\alpha + 1)
\biggl(
1
\alpha + 2
\biggr) 1 - 1
q
\times
\times
\Biggl\{ \biggl[
m\scrC 0(s, \alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1
\alpha - s+ 2
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr] 1
q
+
\biggl[
m
\alpha - s+ 2
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \scrC 0(s, \alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr] 1
q
\Biggr\}
,
where
\scrC 0(s, \alpha , t) =
1\int
0
(1 - t)\alpha +1(1 + t) - sdt.
Corollary 3.2. In Theorem 3.1, if we take h1(t) = h2(t) = t(1 - t), then we have the following
inequality for generalized relative semi-(m, tgs)-preinvex functions:
\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq 2\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)
2+ 2
q
\biggl(
1
\alpha + 2
\biggr) 1 - 1
q
\times
\times
\biggl[
n
\alpha + 3
+
1
(\alpha + 3)(\alpha + 4)
\biggr] 1
q \Bigl(
m
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr) 1
q
.
Corollary 3.3. In Theorem 3.1, if we take h1(t) =
\surd
1 - t
2
\surd
t
and h2(t) =
\surd
t
2
\surd
1 - t
, then we have:
(1) for n > 1, the following inequality for generalized relative semi-m-MT -preinvex functions
holds: \bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
1
\alpha + 2
\biggr) 1 - 1
q
\times
\times
\left\{
\left[ mn
1
2
2F1
\biggl[
- 1
2
, 1;\alpha +
5
2
; - 1
n
\biggr]
2\alpha + 3
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + n - 1
2
2F1
\biggl[
1
2
, 1;\alpha +
7
2
; - 1
n
\biggr]
2\alpha + 5
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
+
\left[ mn - 1
2
2F1
\biggl[
1
2
, 1;\alpha +
7
2
; - 1
n
\biggr]
2\alpha + 5
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + n
1
2
2F1
\biggl[
- 1
2
, 1;\alpha +
5
2
; - 1
n
\biggr]
2\alpha + 3
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} ,
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1640 T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI
(2) for n = 1, the following inequality for generalized relative semi-m-MT -preinvex functions
holds:
1
2(\alpha + 1)
\bigm| \bigm| \bigm| G(\alpha ; 1,m, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
2
3+ 1
q (\alpha + 1)
\biggl(
1
\alpha + 2
\biggr) 1 - 1
q
\times
\times
\biggl\{ \Bigl[
m\scrC 1(\alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \scrC 2(\alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr] 1
q
+
\Bigl[
m\scrC 2(\alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \scrC 1(\alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr] 1
q
\biggr\}
,
where \scrC 1(\alpha , t) =
\int 1
0
(1 - t)\alpha +
1
2 (1 + t)
1
2dt and \scrC 2(\alpha , t) =
\int 1
0
(1 - t)\alpha +
3
2 (1 + t) -
1
2dt.
Now, we are ready to state the second theorem in this section.
Theorem 3.2. If | f \prime \prime | q for q > 1 is generalized relative semi-(m,h1, h2)-preinvex with
1
q
+
1
p
=
= 1, then the following inequality for Riemann – Liouville fractional integrals along with h1, h2 :
[0, 1] \rightarrow \BbbR 0, \alpha > 0 and n \in \BbbN + exists:
\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
1
p(\alpha + 1) + 1
\biggr) 1
p
\times
\times
\left\{
\left[ 1\int
0
mh1
\biggl(
1 - t
n+ 1
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1\int
0
h2
\biggl(
1 - t
n+ 1
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
+
\left[ 1\int
0
mh1
\biggl(
n+ t
n+ 1
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1\int
0
h2
\biggl(
n+ t
n+ 1
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} . (3.2)
Proof. Using given hypothesis, Lemma 2.1 and Hölder’s inequality, we have\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq
\leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\left( 1\int
0
(1 - t)(\alpha +1)pdt
\right)
1
p
\left( 1\int
0
\bigm| \bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| \bigm| q dt
\right)
1
q
+
+
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\left( 1\int
0
(1 - t)(\alpha +1)pdt
\right)
1
p
\left( 1\int
0
\bigm| \bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
n+ t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| \bigm| q dt
\right)
1
q
\leq
\leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
1
p(\alpha + 1) + 1
\biggr) 1
p
\times
\times
\left\{
\left[ 1\int
0
mh1
\biggl(
1 - t
n+ 1
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1\int
0
h2
\biggl(
1 - t
n+ 1
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES . . . 1641
+
\left[ 1\int
0
mh1
\biggl(
n+ t
n+ 1
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1\int
0
h2
\biggl(
n+ t
n+ 1
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} ,
which completes the proof.
Remark 3.2. In Theorem 3.2, if we take h1(t) = (1 - t)s, h2(t) = ts and \varphi is an identity
mapping along with \eta (\varphi (b), \varphi (a),m) = \varphi (b) - m\varphi (a), m = 1, then we obtain Theorem 2.2 in [14],
specially, for n = 1, we get Theorem 3 in [15].
Corollary 3.4. In Theorem 3.2, if we put h1(t) = h2(t) = t(1 - t), then we have the following
inequality for generalized relative semi-(m, tgs)-preinvex functions:\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq 2\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)
2+ 2
q
\biggl(
1
p(\alpha + 1) + 1
\biggr) 1
p
\biggl(
3n+ 1
6
\biggr) 1
q
\times
\times
\Bigl(
m
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr) 1
q
.
Corollary 3.5. In Theorem 3.2, if we put h1(t) =
\surd
1 - t
2
\surd
t
and h2(t) =
\surd
t
2
\surd
1 - t
, then we have:
(1) for n > 1, the following inequality for generalized relative semi-m-MT -preinvex functions
holds: \bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
1
p(\alpha + 1) + 1
\biggr) 1
p
\times
\times
\Biggl\{ \biggl(
mn
1
2
2F1
\biggl[
- 1
2
, 1;
3
2
; - 1
n
\biggr] \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1
3
n - 1
2
2F1
\biggl[
1
2
, 1;
5
2
; - 1
n
\biggr] \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) 1
q
+
+
\biggl(
1
3
mn - 1
2
2F1
\biggl[
1
2
, 1;
5
2
; - 1
n
\biggr] \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + n
1
2
2F1
\biggl[
- 1
2
, 1;
3
2
; - 1
n
\biggr] \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) 1
q
\Biggr\}
,
(2) for n = 1, the following inequality for generalized relative semi-m-MT -preinvex functions
holds:
1
2(\alpha + 1)
\bigm| \bigm| \bigm| G(\alpha ; 1,m, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq
\leq \eta 2(\varphi (b), \varphi (a),m)
2
3+ 1
q (\alpha + 1)
\biggl(
1
p(\alpha + 1) + 1
\biggr) 1
p
\biggl\{ \Bigl[
m
\Bigl( \pi
2
+ 1
\Bigr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \Bigl( \pi
2
- 1
\Bigr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr] 1
q
+
+
\Bigl[
m
\Bigl( \pi
2
- 1
\Bigr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \Bigl( \pi
2
+ 1
\Bigr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr] 1
q
\biggr\}
.
Now, we are ready to state the third theorem in this section.
Theorem 3.3. Under the assumptions of Thereom 3.2, then the following inequality for Rie-
mann – Liouville fractional integrals with \alpha > 0 and n \in \BbbN + holds:\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
q - 1
(\alpha + 1)(q - p) + q - 1
\biggr) q - 1
q
\times
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1642 T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI
\times
\left\{
\left[ 1\int
0
(1 - t)p(\alpha +1)
\biggl(
mh1
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
\left[ 1\int
0
(1 - t)p(\alpha +1)
\biggl(
mh1
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
\right\} . (3.3)
Proof. Using given hypothesis, Lemma 2.1 and Hölder’s inequality, we have
\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\left( 1\int
0
\Bigl[
(1 - t)(\alpha +1)
\Bigr] q - p
q - 1
dt
\right)
q - 1
q
\times
\times
\left( 1\int
0
\bigl[
(1 - t)\alpha +1
\bigr] p \bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| qdt
\right)
1
q
+
+
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\left( 1\int
0
\Bigl[
(1 - t)(\alpha +1)
\Bigr] q - p
q - 1
dt
\right)
q - 1
q
\times
\times
\left( 1\int
0
\bigl[
(1 - t)\alpha +1
\bigr] p \bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
n+ t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| qdt
\right)
1
q
\leq
\leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
q - 1
(\alpha + 1)(q - p) + q - 1
\biggr) q - 1
q
\times
\times
\left\{
\left[ 1\int
0
(1 - t)p(\alpha +1)
\biggl(
mh1
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
\left[ 1\int
0
(1 - t)p(\alpha +1)
\biggl(
mh1
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
\right\} ,
which completes the proof.
Corollary 3.6. In Theorem 3.3, if we put h1(t) = (1 - t)s and h2(t) = ts, then we have:
(1) for n > 1, the following inequality for generalized relative semi-(m, s)-Breckner-preinvex
functions holds:
\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)
2+ s
q
\biggl(
q - 1
(\alpha + 1)(q - p) + q - 1
\biggr) q - 1
q
\times
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES . . . 1643
\times
\left\{
\left[ mns
2F1
\biggl[
- s, 1; p(\alpha + 1) + 2; - 1
n
\biggr]
p(\alpha + 1) + 1
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1
p(\alpha + 1) + s+ 1
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
+
\left[ m
p(\alpha + 1) + s+ 1
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + ns
2F1
\biggl[
- s, 1; p(\alpha + 1) + 2; - 1
n
\biggr]
p(\alpha + 1) + 1
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} ,
(2) for n = 1, the following inequality for generalized relative semi-(m, s)-Breckner-preinvex
functions holds:
1
2(\alpha + 1)
\bigm| \bigm| \bigm| G(\alpha ; 1,m, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
2
3+ s
q (\alpha + 1)
\biggl(
q - 1
(\alpha + 1)(q - p) + q - 1
\biggr) q - 1
q
\times
\times
\Biggl\{ \biggl[
m\scrC 3(p, s, \alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1
p(\alpha + 1) + s+ 1
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr] 1
q
+
+
\biggl[
m
p(\alpha + 1) + s+ 1
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \scrC 3(p, s, \alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr] 1
q
\Biggr\}
,
where
\scrC 3(p, s, \alpha , t) =
1\int
0
(1 - t)p(\alpha +1)(1 + t)sdt.
Corollary 3.7. In Theorem 3.3, if we put h1(t) = h2(t) = t(1 - t), then we have the following
inequality for generalized relative semi-(m, tgs)-preinvex functions:\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq
\leq 2\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)
2+ 2
q
\biggl(
q - 1
(\alpha + 1)(q - p) + q - 1
\biggr) q - 1
q
\times
\times
\biggl[
n
p(\alpha + 1) + 2
+
1
[p(\alpha + 1) + 2] [p(\alpha + 1) + 3]
\biggr] 1
q \Bigl(
m
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr) 1
q
.
Corollary 3.8. In Theorem 3.3, if we put h1(t) =
\surd
1 - t
2
\surd
t
and h2(t) =
\surd
t
2
\surd
1 - t
, then we have:
(1) for n > 1, the following inequality for generalized relative semi-m-MT -preinvex functions
holds: \bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl[
q - 1
(\alpha + 1)(q - p) + q - 1
\biggr] q - 1
q
\times
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1644 T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI
\times
\left\{
\left[ mn
1
2
2F1
\biggl[
- 1
2
, 1; p(\alpha + 1) +
3
2
; - 1
n
\biggr]
2p(\alpha + 1) + 1
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q+
+
n - 1
2
2F1
\biggl[
1
2
, 1; p(\alpha + 1) +
5
2
; - 1
n
\biggr]
2p(\alpha + 1) + 3
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
+
\left[ mn - 1
2
2F1
\biggl[
1
2
, 1; p(\alpha + 1) +
5
2
; - 1
n
\biggr]
2p(\alpha + 1) + 3
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q+
+
n
1
2
2F1
\biggl[
- 1
2
, 1; p(\alpha + 1) +
3
2
; - 1
n
\biggr]
2p(\alpha + 1) + 1
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} ,
(2) for n = 1, the following inequality for generalized relative semi-m-MT -preinvex functions
holds:
1
2(\alpha + 1)
\bigm| \bigm| \bigm| G(\alpha ; 1,m, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
2
3+ 1
q (\alpha + 1)
\biggl(
q - 1
(\alpha + 1)(q - p) + q - 1
\biggr) q - 1
q
\times
\times
\biggl\{ \Bigl[
m\scrC 4(p, \alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \scrC 5(p, \alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr] 1
q
+
\Bigl[
m\scrC 5(p, \alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \scrC 4(p, \alpha , t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr] 1
q
\biggr\}
,
where
\scrC 4(p, \alpha , t) =
1\int
0
(1 - t)p(\alpha +1) - 1
2 (1 + t)
1
2dt
and
\scrC 5(p, \alpha , t) =
1\int
0
(1 - t)p(\alpha +1)+ 1
2 (1 + t) -
1
2dt.
Finally, we shall prove the following result.
Theorem 3.4. If | f \prime \prime | q for q > 1 is generalized relative semi-(m,h1, h2)-preinvex, then the
following inequality for Riemann – Liouville fractional integrals along with h1, h2 : [0, 1] \rightarrow \BbbR 0,
\alpha > 0, n \in \BbbN + and 0 < \mu , \lambda < \alpha + 2 exists:
\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl[
q - 1
q(\alpha + 2 - \mu ) - 1
\biggr] 1 - 1
q
\times
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES . . . 1645
\times
\left[ 1\int
0
(1 - t)\mu q
\biggl(
mh1
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl[
q - 1
q(\alpha + 2 - \lambda ) - 1
\biggr] 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\lambda q
\biggl(
mh1
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
. (3.4)
Proof. Using given hypothesis, Lemma 2.1 and Hölder’s inequality, we have
\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\left[ 1\int
0
(1 - t)
(\alpha +1 - \mu ) q
q - 1dt
\right] 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\mu q
\bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
1 - t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| qdt
\right]
1
q
+
+
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\left[ 1\int
0
(1 - t)
(\alpha +1 - \lambda ) q
q - 1dt
\right] 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\lambda q
\bigm| \bigm| \bigm| f \prime \prime
\biggl(
m\varphi (a) +
n+ t
n+ 1
\eta (\varphi (b), \varphi (a),m)
\biggr) \bigm| \bigm| \bigm| qdt
\right]
1
q
\leq
\leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl[
q - 1
q(\alpha + 2 - \mu ) - 1
\biggr] 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\mu q
\biggl(
mh1
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl[
q - 1
q(\alpha + 2 - \lambda ) - 1
\biggr] 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\lambda q
\biggl(
mh1
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
n+ t
n+ 1
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
,
which completes the proof.
Corollary 3.9. In Theorem 3.4, if we take h1(t) = (1 - t)s, h2(t) = ts for s \in (0, 1] and \mu = \lambda ,
then we have:
(1) for n > 1, the following inequality for generalized relative semi-(m, s)-Breckner-preinvex
functions exists:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1646 T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI
\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl[
q - 1
q(\alpha + 2 - \mu ) - 1
\biggr] 1 - 1
q
\times
\times
\left\{
\left[ mns
2F1
\biggl[
- s, 1;\mu q + 2; - 1
n
\biggr]
\mu q + 1
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1
\mu q + s+ 1
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
+
\left[ m
\mu q + s+ 1
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + ns
2F1
\biggl[
- s, 1;\mu q + 2; - 1
n
\biggr]
\mu q + 1
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} ,
(2) for n = 1, the following inequality for generalized relative semi-(m, s)-Breckner-preinvex
functions exists:
1
2(\alpha + 1)
\bigm| \bigm| \bigm| G(\alpha ; 1,m, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
2
3+ s
q (\alpha + 1)
\biggl(
q - 1
q(\alpha + 2 - \mu ) - 1
\biggr) 1 - 1
q
\times
\times
\Biggl\{ \biggl[
m\scrC 6(\mu , q, s, t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1
\mu q + s+ 1
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr] 1
q
+
+
\biggl[
m
\mu q + s+ 1
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \scrC 6(\mu , q, s, t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr] 1
q
\Biggr\}
,
where
\scrC 6(\mu , q, s, t) =
1\int
0
(1 - t)\mu q(1 + t)sdt.
Corollary 3.10. In Theorem 3.4, if we take h1(t) = h2(t) = t(1 - t) with \mu = \lambda , then we have
the following inequality for generalized relative semi-(m, tgs)-preinvex functions:\bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq
\leq 2\eta 2(\varphi (b), \varphi (a),m)
(n+ 1)
2+ 2
q
\biggl[
q - 1
q(\alpha + 2 - \mu ) - 1
\biggr] 1 - 1
q
\biggl[
n(\mu q + 3) + 1
(\mu q + 2)(\mu q + 3)
\biggr] 1
q
\times
\times
\Bigl(
m
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr) 1
q
.
Corollary 3.11. In Theorem 3.4, if we take h1(t) =
\surd
1 - t
2
\surd
t
, h2(t) =
\surd
t
2
\surd
1 - t
with \mu = \lambda , then
we have:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES . . . 1647
(1) for n > 1, the following inequality for generalized relative semi-m-MT -preinvex functions
holds: \bigm| \bigm| \bigm| G(\alpha ;n,m,\varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
(n+ 1)2
\biggl(
q - 1
(\alpha + 2 - \mu )q - 1
\biggr) 1 - 1
q
\times
\times
\left\{
\left[ mn
1
2
2F1
\biggl[
- 1
2
, 1;\mu q +
3
2
; - 1
n
\biggr]
2\mu q + 1
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + n - 1
2
2F1
\biggl[
1
2
, 1;\mu q +
5
2
; - 1
n
\biggr]
2\mu q + 3
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
+
\left[ mn - 1
2
2F1
\biggl[
1
2
, 1;\mu q +
5
2
; - 1
n
\biggr]
2\mu q + 3
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + n
1
2
2F1
\biggl[
- 1
2
, 1;\mu q +
3
2
; - 1
n
\biggr]
2\mu q + 1
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} ,
(2) for n = 1, the following inequality for generalized relative semi-m-MT -preinvex functions
holds:
1
2(\alpha + 1)
\bigm| \bigm| \bigm| G(\alpha ; 1,m, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq \eta 2(\varphi (b), \varphi (a),m)
2
3+ 1
q (\alpha + 1)
\biggl(
q - 1
q(\alpha + 2 - \mu ) - 1
\biggr) 1 - 1
q
\times
\times
\biggl\{ \Bigl[
m\scrC 7(\mu , q, t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q+ \scrC 8(\mu , q, t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr] 1
q
+
\Bigl[
m\scrC 8(\mu , q, t)
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + \scrC 7(\mu , q, t)
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\Bigr] 1
q
\biggr\}
,
where
\scrC 7(\mu , q, t) =
1\int
0
(1 - t)\mu q -
1
2 (1 + t)
1
2dt
and
\scrC 8(\mu , q, t) =
1\int
0
(1 - t)\mu q+
1
2 (1 + t) -
1
2dt.
It is worth to mention in this paper that one can calculate the value of \scrC 0(s, \alpha , t), \scrC 1(\alpha , t),
\scrC 2(\alpha , t), . . . using some mathematical software (for example, Maple).
4. Applications to special means. Let a and b be positive real numbers such that a < b, we
recall the following means:
A := A(a, b) =
a+ b
2
, G := G(a, b) =
\surd
ab, H := H(a, b) =
2ab
a+ b
,
Pr := Pr(a, b) =
\biggl(
ar + br
2
\biggr) 1
r
, r \geq 1,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1648 T. S. DU, C. Y. LUO, Z. Z. HUANG, A. KASHURI
I := I(a, b) =
\left\{
1
e
\biggl(
bb
aa
\biggr) 1
b - a
, a \not = b,
a, a = b,
L := L(a, b) =
\left\{
b - a
\mathrm{l}\mathrm{n} b - \mathrm{l}\mathrm{n} a
, a \not = b,
a, a = b,
and
Lp := Lp(a, b) =
\left\{
\biggl[
bp+1 - ap+1
(p+ 1)(b - a)
\biggr] 1
p
, p \not = 0, - 1, and a \not = b,
L(a, b), p = - 1 and a \not = b,
I(a, b), p = 0 and a \not = b,
a, a = b.
Consider the function M := M(\varphi (a), \varphi (b)) : [\varphi (a), \varphi (a) + \eta (\varphi (b), \varphi (a))] \times [\varphi (a), \varphi (a) +
+\eta (\varphi (b), \varphi (a))] \rightarrow \BbbR +, which is one of the above mentioned means and \varphi : I \rightarrow K is a continuous
function.
Replace \eta (\varphi (y), \varphi (x),m) with \eta (\varphi (y), \varphi (x)) and setting \eta (\varphi (y), \varphi (x)) = M(\varphi (x), \varphi (y)) for
m = 1 = n in (3.1), (3.2), (3.3) and (3.4). Therefore one can obtain the following interesting in-
equalities involving the above means as follows:
\bigm| \bigm| \bigm| G(\alpha ; 1, 1, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq M2(\varphi (b), \varphi (a))
4
\biggl(
1
\alpha + 2
\biggr) 1 - 1
q
+
\times
\left\{
\left[ 1\int
0
(1 - t)\alpha +1
\biggl(
h1
\biggl(
1 - t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
\left[ 1\int
0
(1 - t)\alpha +1
\biggl(
h1
\biggl(
1 + t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 + t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
\right\} , (4.1)
\bigm| \bigm| \bigm| G(\alpha ; 1, 1, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq M2(\varphi (b), \varphi (a))
4
\biggl(
1
p(\alpha + 1) + 1
\biggr) 1
p
\times
\times
\left\{
\left[ 1\int
0
h1
\biggl(
1 - t
2
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1\int
0
h2
\biggl(
1 - t
2
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
+
+
\left[ 1\int
0
h1
\biggl(
1 + t
2
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + 1\int
0
h2
\biggl(
1 + t
2
\biggr)
dt
\bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q
\right]
1
q
\right\} , (4.2)
\bigm| \bigm| \bigm| G(\alpha ; 1, 1, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq M2(\varphi (b), \varphi (a))
4
\biggl(
q - 1
(\alpha + 1)(q - p) + q - 1
\biggr) q - 1
q
\times
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
FRACTIONAL TRAPEZIUM-LIKE INEQUALITIES . . . 1649
\times
\left\{
\left[ 1\int
0
(1 - t)p(\alpha +1)
\biggl(
h1
\biggl(
1 - t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
\left[ 1\int
0
(1 - t)p(\alpha +1)
\biggl(
h1
\biggl(
1 + t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 + t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
\right\} , (4.3)
and \bigm| \bigm| \bigm| G(\alpha ; 1, 1, \varphi (a), \varphi (b))(f)
\bigm| \bigm| \bigm| \leq M2(\varphi (b), \varphi (a))
4
\biggl[
q - 1
q(\alpha + 2 - \mu ) - 1
\biggr] 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\mu q
\biggl(
h1
\biggl(
1 - t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 - t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
+
+
M2(\varphi (b), \varphi (a))
(n+ 1)2
\biggl[
q - 1
q(\alpha + 2 - \lambda ) - 1
\biggr] 1 - 1
q
\times
\times
\left[ 1\int
0
(1 - t)\lambda q
\biggl(
h1
\biggl(
1 + t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (a)
\bigm| \bigm| \bigm| q + h2
\biggl(
1 + t
2
\biggr) \bigm| \bigm| \bigm| f \prime \prime (b)
\bigm| \bigm| \bigm| q\biggr) dt
\right]
1
q
, (4.4)
where
G(\alpha ; 1, 1, \varphi (a), \varphi (b))(f) =
2\alpha \Gamma (\alpha + 2)
M\alpha (\varphi (b), \varphi (a))
\Bigl[
J\alpha
(\varphi (a)+ 1
2
M(\varphi (b),\varphi (a))) -
f(\varphi (a))+
+J\alpha
(\varphi (a)+ 1
2
M(\varphi (b),\varphi (a)))+
f (\varphi (a) +M(\varphi (b), \varphi (a)))
\Bigr]
-
- 2(\alpha + 1)f
\biggl(
\varphi (a) +
1
2
M(\varphi (b), \varphi (a))
\biggr)
.
Letting M = A, G, H, Pr, I, L, Lp in (4.1), (4.2), (4.3) and (4.4), we get the inequalities
involving means for a particular choice of twice differentiable generalized relative semi-(h1, h2)-
preinvex function f. Further, applying (4.1), (4.2), (4.3) and (4.4) to generalized relative semi-
s-Breckner-preinvex functions, generalized relative semi-P -preinvex functions, generalized rela-
tive semi-s-Godunova – Levin – Dragomir-preinvex functions, generalized relative semi-tgs-preinvex
functions, generalized relative semi-MT -preinvex functions, respectively, one can obtain various ine-
qualities corresponding to these functions involving means.
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Received 18.01.18
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| id | umjimathkievua-article-6036 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:38Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b3/565011cac787944b78ff743a6ae095b3.pdf |
| spelling | umjimathkievua-article-60362025-03-31T08:49:28Z Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set Du, T. S. Luo , C. Y. Huang , Z. Z. Kashuri , A. Du, T. S. Luo , C. Y. Huang , Z. Z. Kashuri , A. A. Hermite-Hadamard’s inequality Riemann-Liouville fractional integrals relative semi-$(m, h_1, h_2)$-preinvex functions Hermite-Hadamard’s inequality Riemann-Liouville fractional integrals relative semi-$(m, h_1, h_2)$-preinvex functions UDC 517.5 The authors derive a fractional integral equality concerning twice differentiable mappings defined on $m$-invex set. By using this identity, the authors obtain new estimates on generalization of trapezium-like inequalities for mappings whose second order derivatives are generalized relative semi-$(m, h_1, h_2)$-preinvex via fractional integrals. We also discuss some new special cases which can be deduced from our main results. UDC 517.5 Дробовi нерiвностi типу трапецiї з узагальненими вiдносно напiв-$(m, h_1, h_2)$-преiнвексними вiдображеннями на $m$ -iнвекснiй множинi Встановлено дробову інтегральну рівність для двічі диференційовних відображень на $m$-інвексній множині.&nbsp;За допомогою цієї рівності отримано нові оцінки для узагальнених нерівностей типу трапеції для відображень, у яких похідні другого порядку є узагальненими відносно напів-$(m,h_{1},h_{2})$-преінвексними через дробові інтеграли.&nbsp;Також обговорено деякі нові спеціальні випадки, що випливають з отриманих результатів. Institute of Mathematics, NAS of Ukraine 2020-12-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6036 10.37863/umzh.v72i12.6036 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 12 (2020); 1633-1350 Український математичний журнал; Том 72 № 12 (2020); 1633-1350 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6036/8872 |
| spellingShingle | Du, T. S. Luo , C. Y. Huang , Z. Z. Kashuri , A. Du, T. S. Luo , C. Y. Huang , Z. Z. Kashuri , A. A. Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set |
| title | Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set |
| title_alt | Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set |
| title_full | Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set |
| title_fullStr | Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set |
| title_full_unstemmed | Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set |
| title_short | Fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set |
| title_sort | fractional trapezium-like inequalities involving generalized relative semi-$(m, h_1, h_2 )$-preinvex mappings on an $m$-invex set |
| topic_facet | Hermite-Hadamard’s inequality Riemann-Liouville fractional integrals relative semi-$(m h_1 h_2)$-preinvex functions Hermite-Hadamard’s inequality Riemann-Liouville fractional integrals relative semi-$(m h_1 h_2)$-preinvex functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6036 |
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