Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications
UDC 517.5The authors have proved an identity with two parameters for differentiable function with respect to another function via generalized integral operator. By applying the established identity, the generalized trapezium, midpoint and Simpson type integral inequalities have been discovered. It i...
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| author | Kashuri , A. Sarikaya, M. Z. Kashuri , A. Sarikaya, M. Z. |
| author_facet | Kashuri , A. Sarikaya, M. Z. Kashuri , A. Sarikaya, M. Z. |
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| description | UDC 517.5The authors have proved an identity with two parameters for differentiable function with respect to another function via generalized integral operator. By applying the established identity, the generalized trapezium, midpoint and Simpson type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means and new error estimates for the trapezium and midpoint quadrature formula have been analyzed. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities.
  |
| doi_str_mv | 10.37863/umzh.v73i9.805 |
| first_indexed | 2026-03-24T02:04:05Z |
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DOI: 10.37863/umzh.v73i9.805
UDC 517.5
A. Kashuri (Univ. Ismail Qemali, Vlora, Albania),
M. Z. Sarikaya (Düzce Univ., Turkey)
DIFFERENT TYPE PARAMETERIZED INEQUALITIES
FOR PREINVEX FUNCTIONS WITH RESPECT TO ANOTHER FUNCTION
VIA GENERALIZED FRACTIONAL INTEGRAL OPERATORS
AND THEIR APPLICATIONS
РIЗНI ТИПИ ПАРАМЕТРИЗОВАНИХ НЕРIВНОСТЕЙ
ДЛЯ ПРЕIНВЕКСНИХ ФУНКЦIЙ ВIДНОСНО IНШОЇ ФУНКЦIЇ
З ВИКОРИСТАННЯМ УЗАГАЛЬНЕНИХ ДРОБОВИХ
IНТЕГРАЛЬНИХ ОПЕРАТОРIВ ТА ЇХ ЗАСТОСУВАННЯ
The authors have proved an identity with two parameters for differentiable function with respect to another function via
generalized integral operator. By applying the established identity, the generalized trapezium, midpoint and Simpson type
integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for
almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications
of presented results to special means and new error estimates for the trapezium and midpoint quadrature formula have been
analyzed. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities.
Доведено тотожнiсть з двома параметрами для диференцiйовних функцiй вiдносно iншої функцiї з використан-
ням узагальненого iнтегрального оператора. За допомогою цiєї тотожностi отримано iнтегральнi нерiвностi типу
трапецiї, середньої точки та типу Сiмпсона. Зазначено, що результати цього дослiдження охоплюють майже всi
дробовi iнтеграли, якi були вiдкритi упродовж кiлькох останнiх десятилiть. Розглянуто рiзнi спецiальнi випадки.
Також наведено деякi застосування цих результатiв у спецiальних випадках i новi оцiнки похибок для квадратурних
формул типу трапецiї та середньої точки. Iдеї та методи цiєї роботи мають стимулювати подальшi дослiдження в
галузi iнтегральних нерiвностей.
1. Introduction. The following inequality, named Hermite – Hadamard inequality, is one of the most
famous inequalities in the literature for convex functions.
Theorem 1.1. Let f : I \subseteq \BbbR - \rightarrow \BbbR be a convex function and p1, p2 \in I with p1 < p2. Then
the following inequality holds:
f
\biggl(
p1 + p2
2
\biggr)
\leq 1
p2 - p1
p2\int
p1
f(x)dx \leq f(p1) + f(p2)
2
. (1.1)
This inequality (1.1) is also known as trapezium inequality.
The trapezium inequality has remained an area of great interest due to its wide applications in the
field of mathematical analysis. Authors of recent decades have studied (1.1) in the premises of newly
invented definitions due to motivation of convex function. Interested readers see the references [1 – 6,
8, 10, 11, 13, 14, 18, 20 – 25, 27 – 33].
The following inequality is well-known in the literature as Simpson’s inequality.
Theorem 1.2. Let f : [p1, p2] - \rightarrow \BbbR be four time differentiable on the interval (p1, p2) and
having the fourth derivative bounded on (p1, p2) that is \| f (4)\| \infty = \mathrm{s}\mathrm{u}\mathrm{p}x\in (p1,p2) | f
(4)| < \infty . Then
we have
c\bigcirc A. KASHURI, M. Z. SARIKAYA, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9 1181
1182 A. KASHURI, M. Z. SARIKAYA\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p2\int
p1
f(x)dx - p2 - p1
3
\biggl[
f(p1) + f(p2)
2
+ 2f
\biggl(
p1 + p2
2
\biggr) \biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq 1
2880
\| f (4)\| \infty (p2 - p1)
5. (1.2)
Inequality (1.2) gives an error bound for the classical Simpson quadrature formula, which is
one of the most used quadrature formulae in practical applications. In recent years, various genera-
lizations, extensions and variants of such inequalities have been obtained. For other recent results
concerning Simpson type inequalities, see [19, 26].
The aim of this paper is to establish trapezium, midpoint and Simpson type generalized integral
inequalities for preinvex functions with respect to another function, some applications to special
means and new error bounds for midpoint and trapezium quadrature formula. Interestingly, the
special cases of presented results, are fractional integral inequalities. Therefore, it is important to
summarize the study of fractional integrals.
At start, let us recall some mathematical preliminaries and definitions which will be helpful for
further study.
Definition 1.1 [23]. Let f \in L[p1, p2]. Then k-fractional integrals of order \alpha , k > 0 with
p1 \geq 0 are defined by
I\alpha ,k
p+1
f(x) =
1
k\Gamma k(\alpha )
x\int
p1
(x - t)
\alpha
k
- 1f(t)dt, x > p1,
and
I\alpha ,k
p - 2
f(x) =
1
k\Gamma k(\alpha )
p2\int
x
(t - x)
\alpha
k
- 1f(t)dt, p2 > x,
where \Gamma k(\cdot ) is k-gamma function.
For k = 1, k-fractional integrals give Riemann – Liouville integrals. For \alpha = k = 1, k-fractional
integrals give classical integrals.
Definition 1.2 [15, 16]. Let g : [p1, p2] \rightarrow \BbbR be an increasing and positive monotone function
on [p1, p2], having a continuous derivative on (p1, p2). The left-hand side fractional integral of f
with respect to g on [p1, p2] of order \alpha > 0 is defined by
I\alpha ,gp1+f(x) =
1
\Gamma (\alpha )
x\int
p1
g\prime (u)f(u)
[g(x) - g(u)]1 - \alpha
du, x > p1,
provided that the integral exists. The right-hand side fractional integral of f with respect to g on
[p1, p2] of order \alpha > 0 is defined by
I\alpha ,gp2 - f(x) =
1
\Gamma (\alpha )
p2\int
x
g\prime (u)f(u)
[g(u) - g(x)]1 - \alpha
du, x < p2,
provided that the integral exists.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1183
Jleli and Samet in [10] proved the Hadamard type inequality for Riemann – Liouville fractional
integral of a convex function f with respect to another function g. Also in [27], Sarikaya and Ertuğral
defined a function \varphi : [0,\infty ) - \rightarrow [0,\infty ) satisfying the following conditions:
1\int
0
\varphi (t)
t
dt < \infty , (1.3)
1
A
\leq \varphi (s)
\varphi (r)
\leq A for
1
2
\leq s
r
\leq 2, (1.4)
\varphi (r)
r2
\leq B
\varphi (s)
s2
for s \leq r, (1.5)\bigm| \bigm| \bigm| \bigm| \varphi (r)r2
- \varphi (s)
s2
\bigm| \bigm| \bigm| \bigm| \leq C| r - s| \varphi (r)
r2
for
1
2
\leq s
r
\leq 2, (1.6)
where A, B, C > 0 are independent of r, s > 0. If \varphi (r)r\alpha is increasing for some \alpha \geq 0 and
\varphi (r)
r\beta
is decreasing for some \beta \geq 0, then \varphi satisfies (1.3) – (1.6) (see [28]). Therefore, the left- and
right-hand sided generalized integral operators are defined as follows:
p+1
I\varphi f(x) =
x\int
p1
\varphi (x - t)
x - t
f(t)dt, x > p1,
p - 2
I\varphi f(x) =
p2\int
x
\varphi (t - x)
t - x
f(t)dt, x < p2.
The most important feature of generalized integrals is that they produce Riemann – Liouville frac-
tional integrals, k-Riemann – Liouville fractional integrals, Katugampola fractional integrals, con-
formable fractional integrals, Hadamard fractional integrals etc. (see [9, 12, 27]).
Recently, Farid in [7] generalized the above integral by introducing an increasing and positive
monotone function g on [p1, p2], having continuous derivative on (p1, p2). The generalized fractional
integral operator defined by Farid may be given as follows.
Definition 1.3. The left- and right-hand sided generalized fractional integral of a function f
with respect to another function g may be given as follows, respectively:
G\varphi ,g
p1+f(x) =
x\int
p1
\varphi (g(x) - g(u))
g(x) - g(u)
g\prime (u)f(u)du, x > p1, (1.7)
G\varphi ,g
p2 - f(x) =
p2\int
x
\varphi (g(u) - g(x))
g(u) - g(x)
g\prime (u)f(u)du, x < p2. (1.8)
This operator generalizes the various fractional integrals of a function f with respect to another
function g.
The following special cases are focussed in our study.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1184 A. KASHURI, M. Z. SARIKAYA
(i) If we take \varphi (u) = u, then the operator (1.7) and (1.8) reduces to Riemann – Liouville integral
of f with respect to function g:
Igp1+f(x) =
x\int
p1
g\prime (u)f(u)du, x > p1, (1.9)
Igp2 - f(x) =
p2\int
x
g\prime (u)f(u)du, x < p2. (1.10)
If g(u) = u, then (1.9) and (1.10) will reduce to Riemann integral of f.
(ii) If we take \varphi (u) =
u\alpha
\Gamma (\alpha )
, then the operator (1.7) and (1.8) reduces to Riemann – Liouville
fractional integral of f with respect to function g:
I\varphi ,gp1+f(x) =
1
\Gamma (\alpha )
x\int
p1
[g(x) - g(u)]\alpha - 1g\prime (u)f(u)du, x > p1, (1.11)
I\varphi ,gp2 - f(x) =
1
\Gamma (\alpha )
p2\int
x
[g(u) - g(x)]\alpha - 1g\prime (u)f(u)du, x < p2. (1.12)
If g(u) = u, then (1.11) and (1.12) will reduce to left- and right-hand sided Riemann – Liouville
fractional integrals of f, respectively.
(iii) If we take \varphi (u) =
u
\alpha
k
k\Gamma k(\alpha )
, then the operator (1.7) and (1.8) reduces to k-Riemann –
Liouville fractional integral of f with respect to function g:
I\varphi ,gp1+,kf(x) =
1
k\Gamma k(\alpha )
x\int
p1
[g(x) - g(u)]
\alpha
k
- 1g\prime (u)f(u)du, x > p1, (1.13)
I\varphi ,gp2 - ,kf(x) =
1
k\Gamma k(\alpha )
p2\int
x
[g(u) - g(x)]\alpha - 1g\prime (u)f(u)du, x < p2. (1.14)
If g(u) = u, then these operators in (1.13) and (1.14) reduces to k-fractional integral operators given
in [23].
(iv) If we take \varphi g(u) = u(g(p2) - u)\alpha - 1 for \alpha \in (0, 1), then the operator given in (1.7) and
(1.8) reduces to conformable fractional integral operator of f with respect to a function g:
K\alpha ,g
p1 f(x) =
x\int
p1
[g(u)]\alpha - 1 g\prime (u)f(u)du, x > p1. (1.15)
This operator (1.15) generalizes conformable fractional integral operator which was given by Khalil
et al. in [17].
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1185
(v) If we take \varphi (u) =
u
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}( - Au), where A =
1 - \alpha
\alpha
and \alpha \in (0, 1), then the operator
given in (1.7) and (1.8) reduces to fractional integral operator of f with respect to function g with
exponential kernel:
J\alpha ,g
p1+f(x) =
1
\alpha
x\int
p1
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- A(g(x) - g(u))
\bigr)
g\prime (u)f(u)du, x > p1, (1.16)
J\alpha ,g
p2 - f(x) =
1
\alpha
p2\int
x
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- A(g(x) - g(u))
\bigr)
g\prime (u)f(u)du, x < p2. (1.17)
Operators in (1.16) and (1.17) generalizes fractional integral operator with exponential kernel which
was introduced by Kirane and Torebek in [18].
Motivated by the above literatures, the main objective of this paper is to discover in Section 2,
an interesting identity with two parameters in order to study some new bounds regarding trapezium,
midpoint and Simpson type integral inequalities. By using the established identity as an auxiliary
result, some new estimates for trapezium, midpoint and Simpson type integral inequalities via ge-
neralized integrals are obtained. It is pointed out that some new fractional integral inequalities have
been deduced from main results. In Section 3, some applications to special means and new error
estimates for the midpoint and trapezium quadrature formula are given. The ideas and techniques of
this paper may stimulate further research in the field of integral inequalities.
2. Main results. Throughout this study, let P = [mp1,mp1 + \eta (p2,mp1)] be an invex subset
with respect to \eta : P \times P - \rightarrow \BbbR , where p1 < p2 and m \in (0, 1]. Also, for all t \in [0, 1], for brevity,
we define
\Lambda \varphi ,g
m (t) :=
t\int
0
\varphi (g (mp1 + u\eta (p2,mp1)) - g(mp1))
g (mp1 + u\eta (p2,mp1)) - g(mp1)
g\prime (mp1 + u\eta (p2,mp1)) du < \infty
and
\Delta \varphi ,g
m (t) :=
1\int
t
\varphi (g (mp1 + \eta (p2,mp1)) - g (mp1 + u\eta (p2,mp1)))
g (mp1 + \eta (p2,mp1)) - g (mp1 + u\eta (p2,mp1))
\times
\times g\prime (mp1 + u\eta (p2,mp1)) du < \infty ,
where g is an increasing and positive monotone function on P, having continuous derivative on
P \circ = (mp1,mp1 + \eta (p2,mp1)).
For establishing some new results regarding general fractional integrals we need to prove the
following lemma.
Lemma 2.1. Let f : P - \rightarrow \BbbR be a differentiable mapping on P \circ and \gamma 1, \gamma 2 \in \BbbR . If f \prime \in L(P ),
then the following identity for generalized fractional integrals holds:
\gamma 1f(mp1) + \gamma 2f(mp1 + \eta (p2,mp1))
2
+
\left[ \Lambda \varphi ,g
m
\biggl(
1
2
\biggr)
+\Delta \varphi ,g
m
\biggl(
1
2
\biggr)
2
- \gamma 1 + \gamma 2
2
\right] \times
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1186 A. KASHURI, M. Z. SARIKAYA
\times f
\biggl(
mp1 +
\eta (p2,mp1)
2
\biggr)
- 1
2\eta (p2,mp1)
\times
\times
\Biggl[
G\varphi ,g\Bigl(
mp1+
\eta (p2,mp1)
2
\Bigr) +f (mp1 + \eta (p2,mp1)) +G\varphi ,g\Bigl(
mp1+
\eta (p2,mp1)
2
\Bigr) - f(mp1)
\Biggr]
=
=
\eta (p2,mp1)
2
\left\{
1
2\int
0
[\Lambda \varphi ,g
m (t) - \gamma 1] f
\prime (mp1 + t\eta (p2,mp1)dt -
-
1\int
1
2
[\Delta \varphi ,g
m (t) - \gamma 2]f
\prime (mp1 + t\eta (p2,mp1))dt
\right\} .
We denote
Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2) :=
\eta (p2,mp1)
2
\times
\times
\left\{
1
2\int
0
[\Lambda \varphi ,g
m (t) - \gamma 1] f
\prime (mp1 + t\eta (p2,mp1))dt -
-
1\int
1
2
[\Delta \varphi ,g
m (t) - \gamma 2] f
\prime (mp1 + t\eta (p2,mp1))dt
\right\} . (2.1)
Proof. Integrating by parts equation (2.1) and changing the variable of integration, we have
Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2) =
=
\eta (p2,mp1)
2
\left\{
1
2\int
0
\Lambda \varphi ,g
m (t)f \prime (mp1 + t\eta (p2,mp1)) dt - \gamma 1
1
2\int
0
f \prime (mp1 + t\eta (p2,mp1)) dt -
-
1\int
1
2
\Delta \varphi ,g
m (t)f \prime (mp1 + t\eta (p2,mp1)) dt+ \gamma 2
1\int
1
2
f \prime (mp1 + t\eta (p2,mp1)) dt
\right\} =
=
\eta (p2,mp1)
2
\left\{ \Lambda \varphi ,g
m (t)f (mp1 + t\eta (p2,mp1))
\eta (p2,mp1)
\bigm| \bigm| \bigm| \bigm| \bigm|
1
2
0
- 1
\eta (p2,mp1)
\times
\times
1
2\int
0
\varphi (g (mp1 + t\eta (p2,mp1)) - g(mp1))
g (mp1 + t\eta (p2,mp1)) - g(mp1)
g\prime (mp1 + t\eta (p2,mp1)) f (mp1 + t\eta (p2,mp1)) dt -
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1187
- \gamma 1
\eta (p2,mp1)
f (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| 12
0
- \Delta \varphi ,g
m (t)f (mp1 + t\eta (p2,mp1))
\eta (p2,mp1)
\bigm| \bigm| \bigm| \bigm| \bigm|
1
1
2
-
- 1
\eta (p2,mp1)
1\int
1
2
\varphi (g (mp1 + \eta (p2,mp1)) - g (mp1 + t\eta (p2,mp1)))
g (mp1 + \eta (p2,mp1)) - g (mp1 + t\eta (p2,mp1))
\times
\times g\prime (mp1 + t\eta (p2,mp1)) f (mp1 + t\eta (p2,mp1)) dt+
\gamma 2
\eta (p2,mp1)
f (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| 1
1
2
\right\} =
=
\gamma 1f(mp1) + \gamma 2f(mp1 + \eta (p2,mp1))
2
+
\left[ \Lambda \varphi ,g
m
\biggl(
1
2
\biggr)
+\Delta \varphi ,g
m
\biggl(
1
2
\biggr)
2
- \gamma 1 + \gamma 2
2
\right] \times
\times f
\biggl(
mp1 +
\eta (p2,mp1)
2
\biggr)
- 1
2\eta (p2,mp1)
\times
\times
\Biggl[
G\varphi ,g\Bigl(
mp1+
\eta (p2,mp1)
2
\Bigr) +f (mp1 + \eta (p2,mp1)) +G\varphi ,g\Bigl(
mp1+
\eta (p2,mp1)
2
\Bigr) - f(mp1)
\Biggr]
.
Lemma 2.1 is proved.
Remark 2.1. 1. Taking m = 1, \gamma 1 = \gamma 2 = 0, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in
Lemma 2.1, we get the classical midpoint type identity.
2. Taking m = 1, \gamma 1 = \gamma 2 =
1
2
, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in Lemma 2.1,
we get the classical Hermite – Hadamard type identity.
3. Taking m = 1, \gamma 1 = \gamma 2 = 1, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in Lemma 2.1, we
get the new Simpson type identity.
Theorem 2.1. Let f : P - \rightarrow \BbbR be a differentiable mapping on P \circ and 0 \leq \gamma 1, \gamma 2 \leq 1. If
| f \prime | q is preinvex on P for q > 1 and p - 1 + q - 1 = 1, then the following inequality for generalized
fractional integrals holds: \bigm| \bigm| Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2)
\bigm| \bigm| \leq \eta (p2,mp1)
2 q
\surd
8
\times
\times
\Biggl\{
p
\sqrt{}
B\varphi ,g
\Lambda m
(\gamma 1; p)
q
\sqrt{}
3| f \prime (mp1)| q + | f \prime (p2)| q + p
\sqrt{}
B\varphi ,g
\Delta m
(\gamma 2; p)
q
\sqrt{}
| f \prime (mp1)| q + 3| f \prime (p2)| q
\Biggr\}
,
where
B\varphi ,g
\Lambda m
(\gamma 1; p) :=
1
2\int
0
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| pdt, B\varphi ,g
\Delta m
(\gamma 2; p) :=
1\int
1
2
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| pdt.
Proof. From Lemma 2.1, preinvexity of | f \prime | q, Hölder inequality and properties of the modulus,
we have
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1188 A. KASHURI, M. Z. SARIKAYA
\bigm| \bigm| Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2)
\bigm| \bigm| \leq \eta (p2,mp1)
2
\times
\times
\left\{
1
2\int
0
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \prime (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| dt +
+
1\int
1
2
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \prime (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| dt
\right\} \leq
\leq \eta (p2,mp1)
2
\left\{
\left(
1
2\int
0
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| pdt
\right)
1
p
\left(
1
2\int
0
\bigm| \bigm| \bigm| f \prime (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| qdt
\right)
1
q
+
+
\left( 1\int
1
2
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| pdt
\right)
1
p
\left( 1\int
1
2
\bigm| \bigm| \bigm| f \prime (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| qdt
\right)
1
q
\right\} \leq
\leq \eta (p2,mp1)
2
\left\{ p
\sqrt{}
B\varphi ,g
\Lambda m
(\gamma 1; p)
\left(
1
2\int
0
\Bigl[
(1 - t)| f \prime (mp1)| q + t| f \prime (p2)| q
\Bigr]
dt
\right)
1
q
+
+ p
\sqrt{}
B\varphi ,g
\Delta m
(\gamma 2; p)
\left( 1\int
1
2
\Bigl[
(1 - t)| f \prime (mp1)| q + t| f \prime (p2)| q
\Bigr]
dt
\right)
1
q
\right\} =
=
\eta (p2,mp1)
2 q
\surd
8
\times
\times
\Biggl\{
p
\sqrt{}
B\varphi ,g
\Lambda m
(\gamma 1; p)
q
\sqrt{}
3| f \prime (mp1)| q + | f \prime (p2)| q + p
\sqrt{}
B\varphi ,g
\Delta m
(\gamma 2; p)
q
\sqrt{}
| f \prime (mp1)| q + 3| f \prime (p2)| q
\Biggr\}
.
Theorem 2.1 is proved.
We point out some special cases of Theorem 2.1.
Corollary 2.1. Taking p = q = 2 in Theorem 2.1, we get
\bigm| \bigm| Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2)
\bigm| \bigm| \leq \eta (p2,mp1)
4
\surd
2
\times
\times
\Biggl\{ \sqrt{}
B\varphi ,g
\Lambda m
(\gamma 1; 2)
\sqrt{}
3| f \prime (mp1)| 2 + | f \prime (p2)| 2 +
\sqrt{}
B\varphi ,g
\Delta m
(\gamma 2; 2)
\sqrt{}
| f \prime (mp1)| 2 + 3| f \prime (p2)| 2
\Biggr\}
.
Corollary 2.2. Taking | f \prime | \leq K in Theorem 2.1, we get
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1189
\bigm| \bigm| Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2)
\bigm| \bigm| \leq K\eta (p2,mp1)
2 q
\surd
2
\Biggl\{
p
\sqrt{}
B\varphi ,g
\Lambda m
(\gamma 1; p) +
p
\sqrt{}
B\varphi ,g
\Delta m
(\gamma 2; p)
\Biggr\}
.
Corollary 2.3. Taking m = 1, \gamma 1 = \gamma 2 = 0, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in
Theorem 2.1, we get the following midpoint type inequality:\bigm| \bigm| Tf,\Lambda 1,\Delta 1(0, 0; p1, p2)
\bigm| \bigm| \leq
\leq p2 - p1
8 q
\surd
4 p
\surd
p+ 1
\Bigl\{
q
\sqrt{}
| f \prime (p1)| q + 3| f \prime (p2)| q + q
\sqrt{}
3| f \prime (p1)| q + | f \prime (p2)| q
\Bigr\}
.
Corollary 2.4. Taking m = 1, \gamma 1 = \gamma 2 =
1
2
, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in
Theorem 2.1, we obtain the following trapezium type inequality:\bigm| \bigm| \bigm| \bigm| \bigm| Tf,\Lambda 1,\Delta 1
\biggl(
1
2
,
1
2
; p1, p2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq p2 - p1
2 q
\surd
8 p
\sqrt{}
2p+1(p+ 1)
\Bigl\{
q
\sqrt{}
| f \prime (p1)| q + 3| f \prime (p2)| q + q
\sqrt{}
3| f \prime (p1)| q + | f \prime (p2)| q
\Bigr\}
.
Corollary 2.5. Taking m = 1, \gamma 1 = \gamma 2 = 1, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in
Theorem 2.1, we have the following new Simpson type inequality:\bigm| \bigm| Tf,\Lambda 1,\Delta 1(1, 1; p1, p2)
\bigm| \bigm| \leq
\leq p2 - p1
8 q
\surd
4
p
\sqrt{}
2p+1 - 1
p+ 1
\Bigl\{
q
\sqrt{}
| f \prime (p1)| q + 3| f \prime (p2)| q + q
\sqrt{}
3| f \prime (p1)| q + | f \prime (p2)| q
\Bigr\}
.
Corollary 2.6. Taking m = 1, \gamma 1 =
1
6
, \gamma 2 =
5
6
, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in
Theorem 2.1, we get \bigm| \bigm| \bigm| \bigm| \bigm| Tf,\Lambda 1,\Delta 1
\biggl(
1
6
,
5
6
; p1, p2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq p2 - p1
24 q
\surd
4
p
\sqrt{}
3 (2p+1 + 1)
p+ 1
\Bigl\{
q
\sqrt{}
| f \prime (p1)| q + 3| f \prime (p2)| q + q
\sqrt{}
3| f \prime (p1)| q + | f \prime (p2)| q
\Bigr\}
.
Corollary 2.7. Taking \gamma 1 = \gamma 2 = 0 and \varphi (t) = t in Theorem 2.1, we have\bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq 1
2 q
\surd
8 p
\sqrt{}
\eta (p2,mp1)
\times
\times
\Biggl\{
p
\sqrt{}
Bg
1(p)
q
\sqrt{}
3| f \prime (mp1)| q + | f \prime (p2)| q + p
\sqrt{}
Bg
2(p)
q
\sqrt{}
| f \prime (mp1)| q + 3| f \prime (p2)| q
\Biggr\}
,
where
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1190 A. KASHURI, M. Z. SARIKAYA
Bg
1(p) :=
mp1+
\eta (p2,mp1)
2\int
mp1
\bigl[
g(t) - g(mp1)
\bigr] p
dt
and
Bg
2(p) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
\bigl[
g(mp1 + \eta (p2,mp1)) - g(t)
\bigr] p
dt.
Corollary 2.8. Taking \gamma 1 = \gamma 2 = 0 and \varphi (t) =
t\alpha
\Gamma (\alpha )
in Theorem 2.1, we obtain
\bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq 1
2 q
\surd
8 p
\sqrt{}
\eta (p2,mp1)
\times
\times
\Biggl\{
p
\sqrt{}
Bg
3(p, \alpha )
q
\sqrt{}
3| f \prime (mp1)| q + | f \prime (p2)| q + p
\sqrt{}
Bg
4(p, \alpha )
q
\sqrt{}
| f \prime (mp1)| q + 3| f \prime (p2)| q
\Biggr\}
,
where
Bg
3(p, \alpha ) :=
mp1+
\eta (p2,mp1)
2\int
mp1
\bigl[
g(t) - g(mp1)
\bigr] p\alpha
dt
and
Bg
4(p, \alpha ) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
\bigl[
g(mp1 + \eta (p2,mp1)) - g(t)
\bigr] p\alpha
dt.
Corollary 2.9. Taking \gamma 1 = \gamma 2 = 0 and \varphi (t) =
t
\alpha
k
k\Gamma k(\alpha )
in Theorem 2.1, we get
\bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq 1
2 q
\surd
8 p
\sqrt{}
\eta (p2,mp1)
\times
\times
\Biggl\{
p
\sqrt{}
Bg
5(p, \alpha , k)
q
\sqrt{}
3| f \prime (mp1)| q + | f \prime (p2)| q + p
\sqrt{}
Bg
6(p, \alpha , k)
q
\sqrt{}
| f \prime (mp1)| q + 3| f \prime (p2)| q
\Biggr\}
,
where
Bg
5(p, \alpha , k) :=
mp1+
\eta (p2,mp1)
2\int
mp1
\bigl[
g(t) - g(mp1)
\bigr] p\alpha
k dt
and
Bg
6(p, \alpha , k) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
\bigl[
g(mp1 + \eta (p2,mp1)) - g(t)
\bigr] p\alpha
k dt.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1191
Corollary 2.10. Taking \gamma 1 = \gamma 2 = 0 and \varphi g(t) = t(g(mp1+\eta (p2,mp1)) - t)\alpha - 1 in Theorem 2.1,
we have \bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq 1
2 q
\surd
8 p
\sqrt{}
\eta (p2,mp1)
\times
\times
\Biggl\{
p
\sqrt{}
Bg
7(p)
q
\sqrt{}
3| f \prime (mp1)| q + | f \prime (p2)| q + p
\sqrt{}
Bg
8(p, \alpha )
q
\sqrt{}
| f \prime (mp1)| q + 3| f \prime (p2)| q
\Biggr\}
,
where
Bg
7(p) =
mp1+
\eta (p2,mp1)
2\int
mp1
\Biggl\{
g\alpha (mp1 + \eta (p2,mp1)) -
\Bigl[
g(mp1) + g(mp1 + \eta (p2,mp1)) - g(t)
\Bigr] \alpha \Biggr\} p
dt
and
Bg
8(p, \alpha ) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
\Bigl[
g\alpha (mp1 + \eta (p2,mp1)) - g\alpha (t)
\Bigr] p
dt.
Corollary 2.11. Taking \gamma 1 = \gamma 2 = 0 and \varphi (t) =
t
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}( - At), where A =
1 - \alpha
\alpha
, in Theo-
rem 2.1, we obtain \bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq 1
2 q
\surd
8 p
\sqrt{}
\eta (p2,mp1)
\times
\times
\Biggl\{
p
\sqrt{}
Bg
9(p,A)
q
\sqrt{}
3| f \prime (mp1)| q + | f \prime (p2)| q + p
\sqrt{}
Bg
10(p,A)
q
\sqrt{}
| f \prime (mp1)| q + 3| f \prime (p2)| q
\Biggr\}
,
where
Bg
9(p,A) :=
mp1+
\eta (p2,mp1)
2\int
mp1
\Bigl\{
1 - \mathrm{e}\mathrm{x}\mathrm{p}
\bigl[
A (g(mp1) - g(t))
\bigr] \Bigr\} p
dt
and
Bg
10(p,A) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
\Bigl\{
1 - \mathrm{e}\mathrm{x}\mathrm{p}
\bigl[
A (g(t) - g(mp1 + \eta (p2,mp1)))
\bigr] \Bigr\} p
dt.
Theorem 2.2. Let f : P - \rightarrow \BbbR be a differentiable mapping on P \circ and 0 \leq \gamma 1, \gamma 2 \leq 1. If | f \prime | q
is preinvex on P for q \geq 1, then the following inequality for generalized fractional integrals holds:\bigm| \bigm| Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2)
\bigm| \bigm| \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1192 A. KASHURI, M. Z. SARIKAYA
\leq \eta (p2,mp1)
2
\Biggl\{ \Bigl[
B\varphi ,g
\Lambda m
(\gamma 1; 1)
\Bigr] 1 - 1
q q
\sqrt{}
D\varphi ,g
\Lambda m
(\gamma 1)| f \prime (mp1)| q + E\varphi ,g
\Lambda m
(\gamma 1)| f \prime (p2)| q +
+
\Bigl[
B\varphi ,g
\Delta m
(\gamma 2; 1)
\Bigr] 1 - 1
q q
\sqrt{}
F\varphi ,g
\Delta m
(\gamma 2)| f \prime (mp1)| q +H\varphi ,g
\Delta m
(\gamma 2)| f \prime (p2)| q
\Biggr\}
,
where
D\varphi ,g
\Lambda m
(\gamma 1) :=
1
2\int
0
(1 - t)
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| dt, E\varphi ,g
\Lambda m
(\gamma 1) :=
1
2\int
0
t
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| dt,
F\varphi ,g
\Delta m
(\gamma 2) :=
1\int
1
2
(1 - t)
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| dt, H\varphi ,g
\Delta m
(\gamma 2) :=
1\int
1
2
t
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| dt,
and B\varphi ,g
\Lambda m
(\gamma 1; 1), B
\varphi ,g
\Delta m
(\gamma 2; 1) are defined as in Theorem 2.1.
Proof. From Lemma 2.1, preinvexity of | f \prime | q, power mean inequality and properties of the
modulus, we have \bigm| \bigm| Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2)
\bigm| \bigm| \leq \eta (p2,mp1)
2
\times
\times
\left\{
1
2\int
0
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \prime (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| dt+
+
1\int
1
2
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \prime (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| dt
\right\} \leq
\leq \eta (p2,mp1)
2
\times
\times
\left\{
\left(
1
2\int
0
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| dt
\right)
1 - 1
q
\left(
1
2\int
0
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \prime (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| qdt
\right)
1
q
+
+
\left( 1\int
1
2
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| dt
\right)
1 - 1
q
\left( 1\int
1
2
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f \prime (mp1 + t\eta (p2,mp1))
\bigm| \bigm| \bigm| qdt
\right)
1
q
\right\} \leq
\leq \eta (p2,mp1)
2
\left\{
\Bigl[
B\varphi ,g
\Lambda m
(\gamma 1; 1)
\Bigr] 1 - 1
q
\left(
1
2\int
0
\bigm| \bigm| \bigm| \Lambda \varphi ,g
m (t) - \gamma 1
\bigm| \bigm| \bigm| \Bigl[ (1 - t)| f \prime (mp1)| q + t| f \prime (p2)| q
\Bigr]
dt
\right)
1
q
+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1193
+
\Bigl[
B\varphi ,g
\Delta m
(\gamma 2; 1)
\Bigr] 1 - 1
q
\left( 1\int
1
2
\bigm| \bigm| \bigm| \Delta \varphi ,g
m (t) - \gamma 2
\bigm| \bigm| \bigm| \Bigl[ (1 - t)| f \prime (mp1)| q + t| f \prime (p2)| q
\Bigr]
dt
\right)
1
q
\right\} =
=
\eta (p2,mp1)
2
\Biggl\{ \Bigl[
B\varphi ,g
\Lambda m
(\gamma 1; 1)
\Bigr] 1 - 1
q q
\sqrt{}
D\varphi ,g
\Lambda m
(\gamma 1)| f \prime (mp1)| q + E\varphi ,g
\Lambda m
(\gamma 1)| f \prime (p2)| q +
+
\Bigl[
B\varphi ,g
\Delta m
(\gamma 2; 1)
\Bigr] 1 - 1
q q
\sqrt{}
F\varphi ,g
\Delta m
(\gamma 2)| f \prime (mp1)| q +H\varphi ,g
\Delta m
(\gamma 2)| f \prime (p2)| q
\Biggr\}
.
Theorem 2.2 is proved.
We point out some special cases of Theorem 2.2.
Corollary 2.12. Taking q = 1 in Theorem 2.2, we get\bigm| \bigm| Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2)
\bigm| \bigm| \leq
\leq \eta (p2,mp1)
2
\Biggl\{ \Bigl[
D\varphi ,g
\Lambda m
(\gamma 1) + F\varphi ,g
\Delta m
(\gamma 2)
\Bigr]
| f \prime (mp1)| +
\Bigl[
E\varphi ,g
\Lambda m
(\gamma 1) +H\varphi ,g
\Delta m
(\gamma 2)
\Bigr]
| f \prime (p2)|
\Biggr\}
.
Corollary 2.13. Taking | f \prime | \leq K in Theorem 2.2, we have\bigm| \bigm| Tf,\Lambda \varphi ,g
m ,\Delta \varphi ,g
m
(\gamma 1, \gamma 2; p1, p2)
\bigm| \bigm| \leq K\eta (p2,mp1)
2
\times
\times
\Biggl\{ \Bigl[
B\varphi ,g
\Lambda m
(\gamma 1; 1)
\Bigr] 1 - 1
q q
\sqrt{}
D\varphi ,g
\Lambda m
(\gamma 1) + E\varphi ,g
\Lambda m
(\gamma 1) +
\Bigl[
B\varphi ,g
\Delta m
(\gamma 2; 1)
\Bigr] 1 - 1
q q
\sqrt{}
F\varphi ,g
\Delta m
(\gamma 2) +H\varphi ,g
\Delta m
(\gamma 2)
\Biggr\}
.
Corollary 2.14. Taking m = 1, \gamma 1 = \gamma 2 = 0, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in
Theorem 2.2, we obtain the following midpoint type inequality:\bigm| \bigm| Tf,\Lambda 1,\Delta 1(0, 0; p1, p2)
\bigm| \bigm| \leq
\leq p2 - p1
16 q
\surd
3
\Bigl\{
q
\sqrt{}
| f \prime (p1)| q + 2| f \prime (p2)| q + q
\sqrt{}
2| f \prime (p1)| q + | f \prime (p2)| q
\Bigr\}
.
Corollary 2.15. Taking m = 1, \gamma 1 = \gamma 2 =
1
2
, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in
Theorem 2.2, we get the following trapezium type inequality:\bigm| \bigm| \bigm| \bigm| \bigm| Tf,\Lambda 1,\Delta 1
\biggl(
1
2
,
1
2
; p1, p2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq p2 - p1
144 q
\surd
6
\Bigl\{
q
\sqrt{}
| f \prime (p1)| q + 5| f \prime (p2)| q + q
\sqrt{}
5| f \prime (p1)| q + | f \prime (p2)| q
\Bigr\}
.
Corollary 2.16. Taking m = 1, \gamma 1 = \gamma 2 = 1, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t in
Theorem 2.2, we have the following new Simpson type inequality:\bigm| \bigm| Tf,\Lambda 1,\Delta 1(1, 1; p1, p2)
\bigm| \bigm| \leq
\leq 3(p2 - p1)
16 q
\surd
9
\Bigl\{
q
\sqrt{}
2| f \prime (p1)| q + 7| f \prime (p2)| q + q
\sqrt{}
7| f \prime (p1)| q + 2| f \prime (p2)| q
\Bigr\}
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1194 A. KASHURI, M. Z. SARIKAYA
Corollary 2.17. Taking m = 1, \gamma 1 =
1
6
, \gamma 2 =
5
6
, \eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t
in Theorem 2.2, we get \bigm| \bigm| \bigm| \bigm| \bigm| Tf,\Lambda 1,\Delta 1
\biggl(
1
6
,
5
6
; p1, p2
\biggr) \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\biggl(
5
72
\biggr) 1 - 1
q p2 - p1
2
\Biggl\{
q
\sqrt{}
\theta 1| f \prime (p1)| q + \theta 2| f \prime (p2)| q + q
\sqrt{}
\theta 3| f \prime (p1)| q + \theta 4| f \prime (p2)| q
\Biggr\}
,
where \theta 1 =
51
1944
+
1
48
, \theta 2 =
29
1296
, \theta 3 =
325
648
and \theta 4 =
125
648
- 7
48
.
Corollary 2.18. Taking \gamma 1 = \gamma 2 = 0 and \varphi (t) = t in Theorem 2.2, we have\bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq
\leq 1
2\eta
q+1
q (p2,mp1)
\Biggl\{ \bigl[
Bg
1(1)
\bigr] 1 - 1
q q
\sqrt{} \Bigl[
Bg
1(1)\eta (p2,mp1) - Cg
1
\Bigr]
| f \prime (mp1)| q + Cg
1 | f \prime (p2)| q +
+
\bigl[
Bg
2(1)
\bigr] 1 - 1
q q
\sqrt{} \Bigl[
Bg
2(1)\eta (p2,mp1) - Eg
1
\Bigr]
| f \prime (mp1)| q + Eg
1 | f \prime (p2)| q
\Biggr\}
,
where
Cg
1 :=
mp1+
\eta (p2,mp1)
2\int
mp1
(t - mp1)(g(t) - g(mp1))dt,
Eg
1 :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
(t - mp1)(g(mp1 + \eta (p2,mp1)) - g(t))dt,
and Bg
1(1), B
g
2(1) are defined as in Corollary 2.7 for value p = 1.
Corollary 2.19. Taking \gamma 1 = \gamma 2 = 0 and \varphi (t) =
t\alpha
\Gamma (\alpha )
in Theorem 2.2, we obtain
\bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq 1
2\eta
q+1
q (p2,mp1)
\times
\times
\Biggl\{ \Bigl[
Bg
3(1, \alpha )
\Bigr] 1 - 1
q q
\sqrt{} \Bigl[
Bg
3(1, \alpha )\eta (p2,mp1) - Cg
1 (\alpha )
\Bigr]
| f \prime (mp1)| q + Cg
1 (\alpha )| f \prime (p2)| q +
+
\Bigl[
Bg
4(1, \alpha )
\Bigr] 1 - 1
q q
\sqrt{} \Bigl[
Bg
4(1, \alpha )\eta (p2,mp1) - Eg
1(\alpha )
\Bigr]
| f \prime (mp1)| q + Eg
1(\alpha )| f \prime (p2)| q
\Biggr\}
,
where
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1195
Cg
1 (\alpha ) :=
mp1+
\eta (p2,mp1)
2\int
mp1
(t - mp1)(g(t) - g(mp1))
\alpha dt,
Eg
1(\alpha ) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
(t - mp1)(g(mp1 + \eta (p2,mp1)) - g(t))\alpha dt,
and Bg
3(1, \alpha ), B
g
4(1, \alpha ) are defined as in Corollary 2.8 for value p = 1.
Corollary 2.20. Taking \gamma 1 = \gamma 2 = 0 and \varphi (t) =
t
\alpha
k
k\Gamma k(\alpha )
in Theorem 2.2, we get
\bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq 1
2\eta
q+1
q (p2,mp1)
\times
\times
\Biggl\{ \Bigl[
Bg
5(1, \alpha , k)
\Bigr] 1 - 1
q q
\sqrt{} \Bigl[
Bg
5(1, \alpha , k)\eta (p2,mp1) - Cg
1 (\alpha , k)
\Bigr]
| f \prime (mp1)| q + Cg
1 (\alpha , k)| f \prime (p2)| q +
+
\Bigl[
Bg
6(1, \alpha , k)
\Bigr] 1 - 1
q q
\sqrt{} \Bigl[
Bg
6(1, \alpha , k)\eta (p2,mp1) - Eg
1(\alpha , k)
\Bigr]
| f \prime (mp1)| q + Eg
1(\alpha , k)| f \prime (p2)| q
\Biggr\}
,
where
Cg
1 (\alpha , k) :=
mp1+
\eta (p2,mp1)
2\int
mp1
(t - mp1)(g(t) - g(mp1))
\alpha
k dt,
Eg
1(\alpha , k) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
(t - mp1)(g(mp1 + \eta (p2,mp1)) - g(t))
\alpha
k dt,
and Bg
5(1, \alpha , k), B
g
6(1, \alpha , k) are defined as in Corollary 2.9 for value p = 1.
Corollary 2.21. Taking \varphi g(t) = t(g(mp1 + \eta (p2,mp1)) - t)\alpha - 1 in Theorem 2.2, we have\bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq
\leq 1
2\eta
q+1
q (p2,mp1)
\Biggl\{ \Bigl[
Bg
7(1)
\Bigr] 1 - 1
q q
\sqrt{} \Bigl[
Bg
7(1)\eta (p2,mp1) - Lg
1
\Bigr]
| f \prime (mp1)| q + Lg
1| f \prime (p2)| q +
+
\Bigl[
Bg
8(1, \alpha )
\Bigr] 1 - 1
q q
\sqrt{} \Bigl[
Bg
8(1, \alpha )\eta (p2,mp1) - Lg
2(\alpha )
\Bigr]
| f \prime (mp1)| q + Lg
2(\alpha )| f \prime (p2)| q
\Biggr\}
,
where
Lg
1(\alpha ) =
mp1+
\eta (p2,mp1)
2\int
mp1
(t - mp1) \times
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1196 A. KASHURI, M. Z. SARIKAYA
\times
\bigl[
g\alpha (mp1 + \eta (p2,mp1)) - (g(mp1) + g(mp1 + \eta (p2,mp1)) - g(t))\alpha
\bigr]
dt,
Lg
2(\alpha ) =
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
(t - mp1)
\bigl[
g\alpha (mp1 + \eta (p2,mp1)) - g\alpha (t)
\bigr]
dt,
and Bg
7(1), B
g
8(1, \alpha ) are defined as in Corollary 2.10 for value p = 1.
Corollary 2.22. Taking \gamma 1 = \gamma 2 = 0 and \varphi (t) =
t
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}( - At), where A =
1 - \alpha
\alpha
, in Theo-
rem 2.2, we obtain \bigm| \bigm| Tf,\Lambda g
m,\Delta g
m
(0, 0, p1, p2)
\bigm| \bigm| \leq 1
2(1 - \alpha )\eta
q+1
q (p2,mp1)
\times
\times
\Biggl\{ \bigl[
Bg
9(1, A)
\bigr] 1 - 1
q q
\sqrt{}
Lg
3(A)| f \prime (mp1)| q + Lg
4(A)| f \prime (p2)| q +
+
\bigl[
Bg
10(1, A)
\bigr] 1 - 1
q q
\sqrt{}
Lg
5(A)| f \prime (mp1)| q + Lg
6(A)| f \prime (p2)| q
\Biggr\}
,
where
Lg
3(A) :=
mp1+
\eta (p2,mp1)
2\int
mp1
(mp1 + \eta (p2,mp1) - t)
\Bigl\{
1 - \mathrm{e}\mathrm{x}\mathrm{p}
\bigl[
A (g(mp1) - g(t))
\bigr] \Bigr\}
dt,
Lg
4(A) :=
mp1+
\eta (p2,mp1)
2\int
mp1
(t - mp1)
\Bigl\{
1 - \mathrm{e}\mathrm{x}\mathrm{p}
\bigl[
A (g(mp1) - g(t))
\bigr] \Bigr\}
dt,
Lg
5(A) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
(mp1 + \eta (p2,mp1) - t) \times
\times
\Bigl\{
1 - \mathrm{e}\mathrm{x}\mathrm{p}
\bigl[
A (g(t) - g(mp1 + \eta (p2,mp1)))
\bigr] \Bigr\}
dt,
Lg
6(A) :=
mp1+\eta (p2,mp1)\int
mp1+
\eta (p2,mp1)
2
(t - mp1)
\Bigl\{
1 - \mathrm{e}\mathrm{x}\mathrm{p}
\bigl[
A (g(t) - g(mp1 + \eta (p2,mp1)))
\bigr] \Bigr\}
dt,
and Bg
9(1, A), B
g
10(1, A) are defined as in Corollary 2.11 for value p = 1.
Remark 2.2. Applying Theorems 2.1 and 2.2 for special values of parameters \gamma 1 and \gamma 2, for
appropriate choices of function g(t) = t; g(t) = \mathrm{l}\mathrm{n} t \forall t > 0; g(t) = et etc., where
\varphi (t) = t,
t\alpha
\Gamma (\alpha )
,
t
\alpha
k
k\Gamma k(\alpha )
,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1197
\varphi g(t) = t(g(p2) - t)\alpha - 1 for \alpha \in (0, 1),
\varphi (t) =
t
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl[ \biggl(
- 1 - \alpha
\alpha
\biggr)
t
\biggr]
for \alpha \in (0, 1),
such that | f \prime | q to be preinvex (or convex in special case), we can deduce some new general fractional
integral inequalities. The details are left to the interested reader.
3. Applications. Consider the following special means for different real numbers p1, p2 and
p1p2 \not = 0:
1) the arithmetic mean:
A := A(p1, p2) =
p1 + p2
2
,
2) the harmonic mean:
H := H(p1, p2) =
2
1
p1
+
1
p2
,
3) the logarithmic mean:
L := L(p1, p2) =
p2 - p1
\mathrm{l}\mathrm{n} | p2| - \mathrm{l}\mathrm{n} | p1|
,
4) the generalized log-mean:
Lr := Lr(p1, p2) =
\Biggl[
pr+1
2 - pr+1
1
(r + 1)(p2 - p1)
\Biggr] 1
r
, r \in \BbbZ \setminus \{ - 1, 0\} .
It is well-known that Lr is monotonic nondecreasing over r \in \BbbZ with L - 1 := L. In particular,
we have the inequality H \leq L \leq A. Now, using the theory results in Section 2, we give some
applications to special means for different real numbers.
Proposition 3.1. Let m \in (0, 1] be a fixed number and p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and
\eta (p2,mp1) > 0. Then, for r \in \BbbN and r \geq 2, where q > 1 and p - 1 + q - 1 = 1, the following
inequality holds: \bigm| \bigm| \bigm| Ar(mp1,mp1 + \eta (p2,mp1)) - Lr
r(mp1,mp1 + \eta (p2,mp1))
\bigm| \bigm| \bigm| \leq
\leq r\eta (p2,mp1)
4 q
\surd
2 p
\surd
p+ 1
\Biggl\{
q
\sqrt{}
A
\bigl(
| mp1| q(r - 1), 3| p2| q(r - 1)
\bigr)
+ q
\sqrt{}
A
\bigl(
3| mp1| q(r - 1), | p2| q(r - 1)
\bigr) \Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 = 0, f(t) = tr and g(t) = \varphi (t) = t in Theorem 2.1, one can obtain the
result immediately.
Proposition 3.2. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for r \in \BbbN
and r \geq 2, where q > 1 and p - 1 + q - 1 = 1, the following inequality holds:\bigm| \bigm| \bigm| A((mp1)
r, (mp1 + \eta (p2,mp1))
r) - Lr
r(mp1,mp1 + \eta (p2,mp1))
\bigm| \bigm| \bigm| \leq
\leq r\eta (p2,mp1)
q
\surd
8 p
\sqrt{}
2p+1(p+ 1)
\Biggl\{
q
\sqrt{}
A
\bigl(
| mp1| q(r - 1), 3| p2| q(r - 1)
\bigr)
+ q
\sqrt{}
A
\bigl(
3| mp1| q(r - 1), | p2| q(r - 1)
\bigr) \Biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1198 A. KASHURI, M. Z. SARIKAYA
Proof. Taking \gamma 1 = \gamma 2 =
1
2
, f(t) = tr and g(t) = \varphi (t) = t in Theorem 2.1, one can obtain
the result immediately.
Proposition 3.3. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for r \in \BbbN
and r \geq 2, where q > 1 and p - 1 + q - 1 = 1, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| A\bigl(
(mp1)
r, (mp1+\eta (p2,mp1))
r
\bigr)
- 1
2
\Biggl[
Ar(mp1,mp1+\eta (p2,mp1))+Lr
r(mp1,mp1+\eta (p2,mp1))
\Biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq r\eta (p2,mp1)
8 q
\surd
2
p
\sqrt{}
2p+1 - 1
p+ 1
\Biggl\{
q
\sqrt{}
A
\bigl(
| mp1| q(r - 1), 3| p2| q(r - 1)
\bigr)
+ q
\sqrt{}
A
\bigl(
3| mp1| q(r - 1), | p2| q(r - 1)
\bigr) \Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 = 1, f(t) = tr and g(t) = \varphi (t) = t in Theorem 2.1, one can obtain the
result immediately.
Proposition 3.4. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for q > 1
and p - 1 + q - 1 = 1, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| 1
A(mp1,mp1 + \eta (p2,mp1))
- 1
L(mp1,mp1 + \eta (p2,mp1))
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq q
\sqrt{}
3
4
\eta (p2,mp1)
4 p
\surd
p+ 1
\Biggl\{
1
q
\sqrt{}
H (| mp1| 2q, 3| p2| 2q)
+
1
q
\sqrt{}
H (3| mp1| 2q, | p2| 2q)
\Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 = 0, f(t) =
1
t
and g(t) = \varphi (t) = t in Theorem 2.1, one can obtain the
result immediately.
Proposition 3.5. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for q > 1
and p - 1 + q - 1 = 1, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| 1
H(mp1,mp1 + \eta (p2,mp1))
- 1
L(mp1,mp1 + \eta (p2,mp1))
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \eta (p2,mp1)
q
\surd
4 p
\sqrt{}
2p+1(p+ 1)
\Biggl\{
1
q
\sqrt{}
H (| mp1| 2q, 3| p2| 2q)
+
1
q
\sqrt{}
H (3| mp1| 2q, | p2| 2q)
\Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 =
1
2
, f(t) =
1
t
and g(t) = \varphi (t) = t in Theorem 2.1, one can obtain
the result immediately.
Proposition 3.6. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for q > 1
and p - 1 + q - 1 = 1, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| 1
H(mp1,mp1 + \eta (p2,mp1))
- 1
2
\Biggl[
1
A(mp1,mp1 + \eta (p2,mp1))
- 1
L(mp1,mp1 + \eta (p2,mp1))
\Biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \eta (p2,mp1)
8 q
\surd
2
p
\sqrt{}
2p+1 - 1
p+ 1
\Biggl\{
1
q
\sqrt{}
H (| mp1| 2q, 3| p2| 2q)
+
1
q
\sqrt{}
H (3| mp1| 2q, | p2| 2q)
\Biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1199
Proof. Taking \gamma 1 = \gamma 2 = 1, f(t) =
1
t
and g(t) = \varphi (t) = t in Theorem 2.1, one can obtain the
result immediately.
Proposition 3.7. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for r \in \BbbN
and r \geq 2, where q \geq 1, the following inequality holds:\bigm| \bigm| \bigm| Ar(mp1,mp1 + \eta (p2,mp1)) - Lr
r(mp1,mp1 + \eta (p2,mp1))
\bigm| \bigm| \bigm| \leq
\leq q
\sqrt{}
2
3
r\eta (p2,mp1)
8
\Biggl\{
q
\sqrt{}
A
\bigl(
| mp1| q(r - 1), 2| p2| q(r - 1)
\bigr)
+ q
\sqrt{}
A
\bigl(
2| mp1| q(r - 1), | p2| q(r - 1)
\bigr) \Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 = 0, f(t) = tr and g(t) = \varphi (t) = t in Theorem 2.2, one can obtain the
result immediately.
Proposition 3.8. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for r \in \BbbN
and r \geq 2, where q \geq 1, the following inequality holds:\bigm| \bigm| \bigm| A((mp1)
r, (mp1 + \eta (p2,mp1))
r) - Lr
r(mp1,mp1 + \eta (p2,mp1))
\bigm| \bigm| \bigm| \leq
\leq r\eta (p2,mp1)
72 q
\surd
3
\Biggl\{
q
\sqrt{}
A
\bigl(
| mp1| q(r - 1), 5| p2| q(r - 1)
\bigr)
+ q
\sqrt{}
A
\bigl(
5| mp1| q(r - 1), | p2| q(r - 1)
\bigr) \Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 =
1
2
, f(t) = tr and g(t) = \varphi (t) = t in Theorem 2.2, one can obtain
the result immediately.
Proposition 3.9. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for r \in \BbbN
and r \geq 2, where q \geq 1, the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| A((mp1)
r, (mp1 + \eta (p2,mp1))
r) -
- 1
2
\Biggl[
Ar(mp1,mp1 + \eta (p2,mp1)) + Lr
r(mp1,mp1 + \eta (p2,mp1))
\Biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq q
\sqrt{}
2
9
3r\eta (p2,mp1)
16
\Biggl\{
q
\sqrt{}
A
\bigl(
2| mp1| q(r - 1), 7| p2| q(r - 1)
\bigr)
+ q
\sqrt{}
A
\bigl(
7| mp1| q(r - 1), 2| p2| q(r - 1)
\bigr) \Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 = 1, f(t) = tr and g(t) = \varphi (t) = t in Theorem 2.2, one can obtain the
result immediately.
Proposition 3.10. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for q \geq 1,
the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| 1
A(mp1,mp1 + \eta (p2,mp1))
- 1
L(mp1,mp1 + \eta (p2,mp1))
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq q
\sqrt{}
2
3
\eta (p2,mp1)
8
\Biggl\{
1
q
\sqrt{}
H (| mp1| 2q, 2| p2| 2q)
+
1
q
\sqrt{}
H (2| mp1| 2q, | p2| 2q)
\Biggr\}
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1200 A. KASHURI, M. Z. SARIKAYA
Proof. Taking \gamma 1 = \gamma 2 = 0, f(t) =
1
t
and g(t) = \varphi (t) = t in Theorem 2.2, one can obtain the
result immediately.
Proposition 3.11. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for q \geq 1,
the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| 1
H(mp1,mp1 + \eta (p2,mp1))
- 1
L(mp1,mp1 + \eta (p2,mp1))
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq \eta (p2,mp1)
72 q
\surd
3
\Biggl\{
1
q
\sqrt{}
H (| mp1| 2q, 5| p2| 2q)
+
1
q
\sqrt{}
H (5| mp1| 2q, | p2| 2q)
\Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 =
1
2
, f(t) =
1
t
and g(t) = \varphi (t) = t in Theorem 2.2, one can obtain
the result immediately.
Proposition 3.12. Let p1, p2 \in \BbbR \setminus \{ 0\} , where p1 < p2 and \eta (p2,mp1) > 0. Then, for q \geq 1,
the following inequality holds:\bigm| \bigm| \bigm| \bigm| \bigm| 1
H(mp1,mp1 + \eta (p2,mp1))
- 1
2
\Biggl[
1
A(mp1,mp1 + \eta (p2,mp1))
- 1
L(mp1,mp1 + \eta (p2,mp1))
\Biggr] \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq q
\sqrt{}
2
9
3\eta (p2,mp1)
16
\Biggl\{
1
q
\sqrt{}
H (2| mp1| 2q, 7| p2| 2q)
+
1
q
\sqrt{}
H (7| mp1| 2q, 2| p2| 2q)
\Biggr\}
.
Proof. Taking \gamma 1 = \gamma 2 = 1, f(t) =
1
t
and g(t) = \varphi (t) = t in Theorem 2.2, one can obtain the
result immediately.
Remark 3.1. Applying our Theorems 2.1 and 2.2 for special values of parameters \gamma 1 and \gamma 2, for
appropriate choices of function g(t) = t; g(t) = \mathrm{l}\mathrm{n} t \forall t > 0; g(t) = et etc., where
\varphi (t) = t,
t\alpha
\Gamma (\alpha )
,
t
\alpha
k
k\Gamma k(\alpha )
,
\varphi g(t) = t(g(p2) - t)\alpha - 1 for \alpha \in (0, 1),
\varphi (t) =
t
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl[ \biggl(
- 1 - \alpha
\alpha
\biggr)
t
\biggr]
for \alpha \in (0, 1),
such that | f \prime | q to be preinvex (or convex in the special case), we can deduce some new general
fractional integral inequalities using above special means. The details are left to the interested reader.
Next, we provide some new error estimates for the midpoint and trapezium quadrature formula.
Let Q be the partition of the points p1 = x0 < x1 < . . . < xk = p2 of the interval [p1, p2]. Let
consider the quadrature formula
p2\int
p1
f(x)dx = M(f,Q) + E(f,Q),
p2\int
p1
f(x)dx = T (f,Q) + E\ast (f,Q),
where
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DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1201
M(f,Q) =
k - 1\sum
i=0
f
\biggl(
xi + xi+1
2
\biggr)
(xi+1 - xi), T (f,Q) =
k - 1\sum
i=0
f(xi) + f(xi+1)
2
(xi+1 - xi)
are the midpoint and trapezium version and E(f,Q), E\ast (f,Q) are denote their associated approxi-
mation errors.
Proposition 3.13. Let f : [p1, p2] - \rightarrow \BbbR be a differentiable function on (p1, p2), where p1 < p2.
If | f \prime | q is convex on [p1, p2] for q > 1 and p - 1 + q - 1 = 1, then the following inequality holds:
\bigm| \bigm| E(f,Q)
\bigm| \bigm| \leq 1
4 q
\surd
4 p
\surd
p+ 1
k - 1\sum
i=0
(xi+1 - xi)
2
\Bigl\{
q
\sqrt{}
| f \prime (xi)| q+3| f \prime (xi+1)| q+ q
\sqrt{}
3| f \prime (xi)| q+| f \prime (xi+1)| q
\Bigr\}
.
Proof. Applying Theorem 2.1 for m = 1, \gamma 1 = \gamma 2 = 0, \eta (p2,mp1) = p2 - mp1 and
g(t) = \varphi (t) = t on the subintervals [xi, xi+1], i = 0, . . . , k - 1, of the partition Q, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| f
\biggl(
xi + xi+1
2
\biggr)
- 1
xi+1 - xi
xi+1\int
xi
f(x)dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq xi+1 - xi
4 q
\surd
4 p
\surd
p+ 1
\Bigl\{
q
\sqrt{}
| f \prime (xi)| q + 3| f \prime (xi+1)| q + q
\sqrt{}
3| f \prime (xi)| q + | f \prime (xi+1)| q
\Bigr\}
. (3.1)
Hence, from (3.1), we get
\bigm| \bigm| E(f,Q)
\bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p2\int
p1
f(x)dx - M(f,Q)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
k - 1\sum
i=0
\left\{
xi+1\int
xi
f(x)dx - f
\biggl(
xi + xi+1
2
\biggr)
(xi+1 - xi)
\right\}
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
k - 1\sum
i=0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\left\{
xi+1\int
xi
f(x)dx - f
\biggl(
xi + xi+1
2
\biggr)
(xi+1 - xi)
\right\}
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
4 q
\surd
4 p
\surd
p+ 1
k - 1\sum
i=0
(xi+1 - xi)
2 \times
\times
\Bigl\{
q
\sqrt{}
| f \prime (xi)| q + 3| f \prime (xi+1)| q + q
\sqrt{}
3| f \prime (xi)| q + | f \prime (xi+1)| q
\Bigr\}
.
Proposition 3.13 is proved.
Proposition 3.14. Let f : [p1, p2] - \rightarrow \BbbR be a differentiable function on (p1, p2), where p1 < p2.
If | f \prime | q is convex on [p1, p2] for q \geq 1, then the following inequality holds:
\bigm| \bigm| E(f,Q)
\bigm| \bigm| \leq 1
8 q
\surd
3
k - 1\sum
i=0
(xi+1 - xi)
2
\Bigl\{
q
\sqrt{}
| f \prime (xi)| q + 2| f \prime (xi+1)| q + q
\sqrt{}
2| f \prime (xi)| q + | f \prime (xi+1)| q
\Bigr\}
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1202 A. KASHURI, M. Z. SARIKAYA
Proof. The proof is analogous as to that of Proposition 3.13 taking m = 1, \gamma 1 = \gamma 2 = 0,
\eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t using Theorem 2.2.
Proposition 3.15. Let f : [p1, p2] - \rightarrow \BbbR be a differentiable function on (p1, p2), where p1 < p2.
If | f \prime | q is convex on [p1, p2] for q > 1 and p - 1 + q - 1 = 1, then the following inequality holds:
\bigm| \bigm| E\ast (f,Q)
\bigm| \bigm| \leq 1
q
\surd
8 p
\sqrt{}
2p+1(p+ 1)
k - 1\sum
i=0
(xi+1 - xi)
2\times
\times
\Bigl\{
q
\sqrt{}
| f \prime (xi)| q + 3| f \prime (xi+1)| q + q
\sqrt{}
3| f \prime (xi)| q + | f \prime (xi+1)| q
\Bigr\}
.
Proof. Applying Theorem 2.1 for m = 1, \gamma 1 = \gamma 2 =
1
2
, \eta (p2,mp1) = p2 - mp1 and
g(t) = \varphi (t) = t on the subintervals [xi, xi+1], i = 0, . . . , k - 1, of the partition Q, we have\bigm| \bigm| \bigm| \bigm| \bigm| f(xi) + f(xi+1)
2
- 1
xi+1 - xi
xi+1\int
xi
f(x)dx
\bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq xi+1 - xi
q
\surd
8 p
\sqrt{}
2p+1(p+ 1)
\Bigl\{
q
\sqrt{}
| f \prime (xi)| q + 3| f \prime (xi+1)| q + q
\sqrt{}
3| f \prime (xi)| q + | f \prime (xi+1)| q
\Bigr\}
. (3.2)
Hence, from (3.2), we get
\bigm| \bigm| E\ast (f,Q)
\bigm| \bigm| =
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
p2\int
p1
f(x)dx - T (f,Q)
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm| \bigm| \bigm| \bigm| \bigm|
k - 1\sum
i=0
\Biggl\{ xi+1\int
xi
f(x)dx - f(xi) + f(xi+1)
2
(xi+1 - xi)
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq
k - 1\sum
i=0
\bigm| \bigm| \bigm| \bigm| \bigm|
\Biggl\{ xi+1\int
xi
f(x)dx - f(xi) + f(xi+1)
2
(xi+1 - xi)
\Biggr\} \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
q
\surd
8 p
\sqrt{}
2p+1(p+ 1)
k - 1\sum
i=0
(xi+1 - xi)
2
\Bigl\{
q
\sqrt{}
| f \prime (xi)| q + 3| f \prime (xi+1)| q + q
\sqrt{}
3| f \prime (xi)| q + | f \prime (xi+1)| q
\Bigr\}
.
Proposition 3.15 is proved.
Proposition 3.16. Let f : [p1, p2] - \rightarrow \BbbR be a differentiable function on (p1, p2), where p1 < p2.
If | f \prime | q is convex on [p1, p2] for q \geq 1, then the following inequality holds:
\bigm| \bigm| E\ast (f,Q)
\bigm| \bigm| \leq 1
72 q
\surd
6
k - 1\sum
i=0
(xi+1 - xi)
2
\Bigl\{
q
\sqrt{}
| f \prime (xi)| q + 5| f \prime (xi+1)| q + q
\sqrt{}
5| f \prime (xi)| q + | f \prime (xi+1)| q
\Bigr\}
.
Proof. The proof is analogous as to that of Proposition 3.15 taking m = 1, \gamma 1 = \gamma 2 =
1
2
,
\eta (p2,mp1) = p2 - mp1 and g(t) = \varphi (t) = t using Theorem 2.2.
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DIFFERENT TYPE PARAMETERIZED INEQUALITIES FOR PREINVEX FUNCTIONS . . . 1203
Remark 3.2. Applying our Theorems 2.1 and 2.2, where m = 1, for special values of parameter
\gamma 1 and \gamma 2, for appropriate choices of function g(t) = t; g(t) = \mathrm{l}\mathrm{n} t \forall t > 0; g(t) = et etc., where
\varphi (t) = t,
t\alpha
\Gamma (\alpha )
,
t
\alpha
k
k\Gamma k(\alpha )
,
\varphi g(t) = t(g(p2) - t)\alpha - 1 for \alpha \in (0, 1),
\varphi (t) =
t
\alpha
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl[ \biggl(
- 1 - \alpha
\alpha
\biggr)
t
\biggr]
for \alpha \in (0, 1),
such that | f \prime | q to be convex, we can deduce some new bounds for the midpoint and trapezium
quadrature formula using above ideas and techniques. The details are left to the interested reader.
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ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
|
| id | umjimathkievua-article-805 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:05Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ef/2c41e4140df1c8f6215a1102f1dc16ef.pdf |
| spelling | umjimathkievua-article-8052025-03-31T08:46:40Z Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications Kashuri , A. Sarikaya, M. Z. Kashuri , A. Sarikaya, M. Z. Trapezium inequality Simpson inequality preinvexity general fractional integrals Trapezium inequality Simpson inequality preinvexity general fractional integrals UDC 517.5The authors have proved an identity with two parameters for differentiable function with respect to another function via generalized integral operator. By applying the established identity, the generalized trapezium, midpoint and Simpson type integral inequalities have been discovered. It is pointed out that the results of this research provide integral inequalities for almost all fractional integrals discovered in recent past decades. Various special cases have been identified. Some applications of presented results to special means and new error estimates for the trapezium and midpoint quadrature formula have been analyzed. The ideas and techniques of this paper may stimulate further research in the field of integral inequalities. &nbsp; УДК 517.5 Рiзнi типи параметризованих нерiвностей для преiнвексних функцiй вiдносно iншої функцiї з використанням узагальнених дробових iнтегральних операторiв та їх застосування Доведено тотожність з двома параметрами для диференційовних функцій відносно іншої функції з використанням узагальненого інтегрального оператора. За допомогою цієї тотожності отримано інтегральні нерівності типу трапеції, середньої точки та типу Сімпсона. Зазначено, що результати цього дослідження охоплюють майже всі дробові інтеграли, які були відкриті упродовж кількох останніх десятиліть. Розглянуто різні спеціальні випадки. Також наведено деякі застосування цих результатів у спеціальних випадках і нові оцінки похибок для квадратурних формул типу трапеції та середньої точки. Ідеї та методи цієї роботи мають стимулювати подальші дослідження в галузі інтегральних нерівностей. Institute of Mathematics, NAS of Ukraine 2021-09-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/805 10.37863/umzh.v73i9.805 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 9 (2021); 1181 - 1204 Український математичний журнал; Том 73 № 9 (2021); 1181 - 1204 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/805/9104 Copyright (c) 2021 Artion Kashuri, Mehmet Zeki Sarikaya |
| spellingShingle | Kashuri , A. Sarikaya, M. Z. Kashuri , A. Sarikaya, M. Z. Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications |
| title | Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications |
| title_alt | Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications |
| title_full | Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications |
| title_fullStr | Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications |
| title_full_unstemmed | Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications |
| title_short | Different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications |
| title_sort | different type parameterized inequalities for preinvex functions with respect to another function via generalized fractional integral operators and their applications |
| topic_facet | Trapezium inequality Simpson inequality preinvexity general fractional integrals Trapezium inequality Simpson inequality preinvexity general fractional integrals |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/805 |
| work_keys_str_mv | AT kashuria differenttypeparameterizedinequalitiesforpreinvexfunctionswithrespecttoanotherfunctionviageneralizedfractionalintegraloperatorsandtheirapplications AT sarikayamz differenttypeparameterizedinequalitiesforpreinvexfunctionswithrespecttoanotherfunctionviageneralizedfractionalintegraloperatorsandtheirapplications AT kashuria differenttypeparameterizedinequalitiesforpreinvexfunctionswithrespecttoanotherfunctionviageneralizedfractionalintegraloperatorsandtheirapplications AT sarikayamz differenttypeparameterizedinequalitiesforpreinvexfunctionswithrespecttoanotherfunctionviageneralizedfractionalintegraloperatorsandtheirapplications |