Inequalities involving new fractional integrals technique via exponentially convex functions
UDC 517.5 We establish some new Hermite–Hadamard type inequalities involving fractional integral operators with the exponential kernel. Meanwhile, we present many useful estimates on these types of new Hermite–Hadamard type inequalities via exponentially convex functions.  
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| author | Rashid, S. Noor, M. A. Noor, K. I. S. Rashid, S. Noor, M. A. Noor, K. I. |
| author_facet | Rashid, S. Noor, M. A. Noor, K. I. S. Rashid, S. Noor, M. A. Noor, K. I. |
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| description | UDC 517.5
We establish some new Hermite–Hadamard type inequalities involving fractional integral operators with the exponential kernel. Meanwhile, we present many useful estimates on these types of new Hermite–Hadamard type inequalities via exponentially convex functions.
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DOI: 10.37863/umzh.v73i9.947
UDC 517.5
S. Rashid (COMSATS Univ., Islamabad, Pakistan),
M. A. Noor (Government College Univ., Faisalabad, Pakistan),
K. I. Noor (COMSATS Univ., Islamabad, Pakistan)
INEQUALITIES INVOLVING NEW FRACTIONAL INTEGRALS TECHNIQUE
VIA EXPONENTIALLY CONVEX FUNCTIONS
НЕРIВНОСТI, ОТРИМАНI ЗА ДОПОМОГОЮ ДРОБОВИХ IНТЕГРАЛIВ
IЗ ВИКОРИСТАННЯМ ЕКСПОНЕНЦIАЛЬНО ОПУКЛИХ ФУНКЦIЙ
We establish some new Hermite – Hadamard type inequalities involving fractional integral operators with the exponential
kernel. Meanwhile, we present many useful estimates on these types of new Hermite – Hadamard type inequalities via
exponentially convex functions.
За допомогою дробових iнтегральних операторiв iз експоненцiальним ядром отримано кiлька нерiвностей типу
Ермiта – Адамара. Серед iншого запропоновано багато корисних оцiнок для цих нових нерiвностей типу Ермiта –
Адамара з використанням експоненцiально опуклих функцiй.
1. Introduction. Fractional calculus can be seen as a generalization of the ordinary differentiation
and integration to an arbitrary non-integer order which has been recognized as one of the most po-
werful tools to describe long-memory processes in the last decades. Many phenomena from physics,
chemistry, mechanics and electricity can be modeled by ordinary differential equations involving
fractional derivatives (see [5, 9 – 12, 20, 21] and the references therein). There were several studies
in the literature that include further properties such as expansion formulas, variational calculus appli-
cations, control theoretical applications, convexity and integral inequalities and Hermite – Hadamard
type inequalities of this new operator and similar operators.
The usefulness of inequalities involving convex functions is realized from the very beginning and
is now widely acknowledged as one of the prime driving forces behind the development of several
modern branches of mathematics and has been given considerable attention. One of the most famous
inequalities for convex functions is Hermite – Hadamard inequality, stated as [8]:
Let \varphi : I \subset \BbbR \rightarrow \BbbR be a convex function on the interval I of real numbers and \varsigma , \eta \in I with
\varsigma < \eta . Then
\varphi
\Bigl( \varsigma + \eta
2
\Bigr)
\leq 1
\eta - \varsigma
\eta \int
\varsigma
\varphi (x)dx \leq \varphi (\varsigma ) + \varphi (\eta )
2
. (1.1)
Both inequalities hold in the reversed direction for \varphi to be concave.
In recent years, numerous generalizations, extensions and variants of Hermite – Hadamard
inequality (1.1) were studied extensively by many researchers and appeared in a number of papers
(see [1, 4, 6, 13 – 16, 18, 22 – 27]).
In [7], Fejér obtained the weighted generalizations of Hermite – Hadamard inequality (1.1) as
follows:
Let \varphi : [\varsigma , \eta ] \rightarrow \BbbR be a convex function. Then the inequality
c\bigcirc S. RASHID, M. A. NOOR, K. I. NOOR, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9 1217
1218 S. RASHID, M. A. NOOR, K. I. NOOR
\varphi
\Bigl( \varsigma + \eta
2
\Bigr) \eta \int
\varsigma
\chi (x)dx \leq 1
\eta - \varsigma
\eta \int
\varsigma
\varphi (x)\chi (x)dx \leq \varphi (\varsigma ) + \varphi (\eta )
2
\eta \int
\varsigma
\chi (x)dx
holds for a nonnegative, integrable function \chi : [\varsigma , \eta ] \rightarrow \BbbR , which is symmetric to
\varsigma + \eta
2
.
Exponentially convex functions have emerged an a significant new class of convex functions,
which have important applications in technology, data science and statistics. The main motivation of
this paper depends on a new identity that has been proved via new fractional integrals operators with
exponential kernel and applied for exponentially convex functions. This identity offers new upper
bounds and estimations of Hadamard type integral inequalities. Some special cases such for \alpha \rightarrow 1
have been discussed, which can be deduced from these results.
To the best of our knowledge, a comprehensive investigation of exponentially convex functions as
fractional integral with exponential kernel in the present paper is new one. The class of exponentially
convex functions was introduced by Antczak [3] and Dragomir [6]. Motivated by these facts, Awan
et al. [4] introduced and investigated another class of convex functions, which is called exponentially
convex function and is significantly different from the class introduced by [3, 6, 19]. The growth
of research on big data analysis and deep learning has recently increased the interest in information
theory involving exponentially convex functions. The smoothness of exponentially convex function
is exploited for statistical learning, sequential prediction and stochastic optimization (see [2, 3, 17]
and the references therein).
It is known [6] that a function \varphi is exponentially convex if and only if \varphi satisfies the inequality
e\varphi (
\varsigma +\eta
2
) \leq 1
\eta - \varsigma
\eta \int
\varsigma
e\varphi (x)dx \leq e\varphi (\varsigma ) + e\varphi (\eta )
2
. (1.2)
The inequality (1.2) is called the Hermite – Hadamard inequality and provides the upper and lower
estimates for the exponential integral (see [22 – 24] and the references therein).
In this paper, we will establish here some new Hermite – Hadamard type inequalities involving
fractional integral with an exponential kernel via exponentially convex functions. Meanwhile, we
present many useful estimates on these types of new Hermite – Hadamard type inequalities for frac-
tional integrals with exponential kernels.
2. Essential preliminaries. We now recall some well-known concepts and basic results, which
are needed in the derivation of our results.
Definition 2.1. A set K \subset \BbbR is said to be convex, if
\tau x+ (1 - \tau )y \in K \forall x, y \in K, \tau \in [0, 1].
Definition 2.2. A function \varphi : K \rightarrow \BbbR is said to be a convex function if and only if
\varphi (\tau x+ (1 - \tau )y) \leq \tau \varphi (x) + (1 - \tau )\varphi (y) \forall x, y \in K, \tau \in [0, 1].
We now consider class of exponentially convex function, which are mainly due to [3, 6].
Definition 2.3 [3, 6]. A positive real-valued function \varphi : K \subseteq \BbbR - \rightarrow (0,\infty ) is said to be
exponentially convex on K, if the inequality
e\varphi (\tau x+(1 - \tau )y) \leq \tau e\varphi (x) + (1 - \tau )e\varphi (y)
holds for x, y \in K and t \in [0, 1].
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
INEQUALITIES INVOLVING NEW FRACTIONAL INTEGRALS TECHNIQUE VIA EXPONENTIALLY . . . 1219
For t =
1
2
, we have Jensen type exponentially convex functions for Definition 2.3:
e\varphi (
x+y
2
) \leq e\varphi (x) + e\varphi (y)
2
.
Exponentially convex functions are used to manipulate for statistical learning, sequential prediction
and stochastic optimization (see [2, 3, 17] and the references therein).
It is known that x \in K is the minimum of the differentiable exponentially convex functions \varphi if
and only if x \in K satisfies the inequality\bigl\langle
(e\varphi (x))\prime , y - x
\bigr\rangle
=
\bigl\langle
\varphi \prime (x)e\varphi (x), y - x
\bigr\rangle
\geq 0 \forall y \in K. (2.1)
The inequality of the type (2.1) is known as the exponentially variational inequality which appears
to be new one.
For formulation, applications and other aspects of variational inequalities, see Noor [13 – 15].
Let us give some basic examples of exponentially convex functions (for details, see [19]).
(i) For every \alpha > 0, the function \varphi (x) = e - \alpha
\surd
x is exponentially convex on (0,\infty ), where
e - \alpha
\surd
x =
\infty \int
0
\alpha
2
\surd
\pi \tau 3
e - \tau (x+( \alpha
2\tau
)2)d\tau , x > 0.
(ii) \varphi (x) = x - \alpha is exponentially convex on (0,\infty ) for any \alpha > 0.
(iii) Let \varsigma , \eta be positive real numbers, I = (0,\infty ) and family F = \{ \varphi \tau : \tau \in I\} of function
defined on C[\varsigma , \eta ] with \varphi t(x) =
e - x
\surd
\tau
( -
\surd
\tau )n
is an exponentially convex on (0,\infty ).
Now, some necessary definitions and mathematical preliminaries of fractional calculus theory are
presented, which are used further in this paper.
Definition 2.4 [10]. Let \varphi \in L1[\varsigma , \eta ]. The fractional integrals \scrI \alpha
\varsigma and \scrI \alpha
\eta of order \alpha \in (0, 1)
are defined as
\scrI \alpha
\varsigma \varphi (x) =
1
\alpha
x\int
\varsigma
e
\alpha - 1
\alpha
(x - u)\varphi (u)du, x > \varsigma , (2.2)
and
\scrI \alpha
\eta \varphi (x) =
1
\alpha
\eta \int
x
e
\alpha - 1
\alpha
(u - x)\varphi (u)du, x < \eta ,
respectively.
If \alpha = 1, then
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 1
\scrI \alpha
\varsigma \varphi (x) =
x\int
\varsigma
\varphi (u)du, \mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 1
\scrI \alpha
\eta \varphi (x) =
\eta \int
x
\varphi (u)du.
Moreover,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1220 S. RASHID, M. A. NOOR, K. I. NOOR
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 0
1
\alpha
e
\alpha - 1
\alpha
(x - u) = \vargamma (x - u),
we conclude that
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 0
\scrI \alpha
\varsigma \varphi (x) = \varphi (x), \mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 1
\scrI \alpha
\eta \varphi (x) = \varphi (x).
Definition 2.5. The left and right Riemann – Liouville fractional integrals J\alpha
\varsigma + and J\alpha
\eta - of order
\alpha \in \BbbR (\alpha > 0) are given by
J\alpha
\varsigma +\varphi (x) =
1
\Gamma (\alpha )
x\int
\varsigma
(x - u)\alpha - 1\varphi (u)du, x > \varsigma ,
and
J\alpha
\eta - \varphi (x) =
1
\Gamma (\alpha )
\eta \int
x
(u - x)\alpha - 1\varphi (u)du, x < \eta ,
respectively. Here \Gamma (\alpha ) is the Euler gamma-function.
In the sequel of the paper, let I \subset \BbbR be a convex set in the finite dimensional Euclidean
space \BbbR n. From now onwards we take I = [\varsigma , \eta ], unless otherwise specified. We henceforth set
\sigma =
1 - \alpha
\alpha
(\eta - \varsigma ).
3. Main results. An important Hermite – Hadamard inequality involving fractional integral with
exponential kernel (with \alpha \in \BbbR , \alpha \geq 0) can be represented as follows.
Theorem 3.1. Let \alpha \in (0, 1) and \varphi : [\varsigma , \eta ] \rightarrow \BbbR be a positive function with 0 \leq \varsigma < \eta and
e\varphi \in L1[\varsigma , \eta ]. If \varphi is a convex function on [\varsigma , \eta ], then the following inequalities for fractional
integrals hold:
e\varphi (
\varsigma +\eta
2
) \leq (1 - \alpha )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[ \scrI \alpha
\varsigma e
\varphi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )
\bigr]
\leq e\varphi (\varsigma ) + e\varphi (\eta )
2
. (3.1)
Proof. Since \varphi is an exponentially convex function on [\varsigma , \eta ], we have, for x, y \in [\varsigma , \eta ] with
\tau =
1
2
,
e\varphi (
x+y
2
) \leq e\varphi (x) + e\varphi (y)
2
,
i.e., with x = \tau \varsigma + (1 - \tau )\eta , y = (1 - \tau )\varsigma + \tau \eta ,
e\varphi (
\varsigma +\eta
2
) \leq e\varphi (\tau \varsigma +(1 - \tau )\eta ) + e\varphi ((1 - \tau )\varsigma +\tau \eta )
2
. (3.2)
Multiplying both sides of the above inequality by e - \sigma \tau and then integrating the resulting inequality
with respect to \tau over [0, 1], we obtain
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr)
\sigma e
\sigma
2
e\varphi (
\varsigma +\eta
2
) \leq
1\int
0
e - \sigma \tau e\varphi (\tau \varsigma +(1 - \tau )\eta )d\tau +
1\int
0
e - \sigma \tau e\varphi ((1 - \tau )\varsigma +\tau \eta )d\tau =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
INEQUALITIES INVOLVING NEW FRACTIONAL INTEGRALS TECHNIQUE VIA EXPONENTIALLY . . . 1221
=
1
\eta - \varsigma
\eta \int
\varsigma
e -
1 - \alpha
\alpha
(\eta - u)ef(u)du+
1
\eta - \varsigma
\eta \int
\varsigma
e -
1 - \alpha
\alpha
(u - \varsigma )ef(u)du =
=
\alpha
\eta - \varsigma
\bigl[
\scrI \alpha
\varsigma e
\varphi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )
\bigr]
and the first inequality in (3.1) is proved.
Because e\varphi is convex function, we have
e\varphi (\tau \varsigma +(1 - \tau )\eta ) + e\varphi ((1 - \tau )\varsigma +\tau \eta ) \leq
\bigl[
e\varphi (\varsigma ) + e\varphi (\eta )
\bigr]
.
Then multiplying both sides of the above inequality by e - \sigma \tau and integrating the resulting inequality
with respect to \tau over [0, 1], we obtain the right-sided inequality in (3.1).
Theorem 3.1 is proved.
Remark 3.1. For \alpha \rightarrow 1, observe that
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 1
(1 - \alpha )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) =
1
\eta - \varsigma
and (3.1) reduces to Theorem 1 in [6].
Remark 3.2. One can follow the same ideas to construct fractional version \scrF (1)
n , \scrF (2)
n , \scrF (3)
n
(see [27]) try to extend to study using exponential kernel for Hermite – Hadamard inequalities in
n variables based on these fundamental results. We shall study such interesting problems in the
forthcoming works.
We now prove the Hermite – Hadamard – Fejér type inequality for exponentially convex function
for new fractional integral operator technique.
Theorem 3.2. Let \varphi : [\varsigma , \eta ] \rightarrow \BbbR be convex and integrable function with \varsigma < \eta . If \chi : [\varsigma , \eta ] \rightarrow \BbbR
is nonnegative, integrable and symmetric with respect to
\varsigma + \eta
2
, that is, \chi (\varsigma + \eta - x) = \chi (x), then
the following fractional integral inequalities hold:
e\varphi (
\varsigma +\eta
2
)
\bigl[
\scrI \alpha
\varsigma e
\chi (\eta ) + \scrI \alpha
\eta e
\chi (\varsigma )
\bigr]
\leq
\bigl[
\scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )
\bigr]
\leq
\leq [e\varphi (\varsigma ) + e\varphi (\eta )]
2
\bigl[
\scrI \alpha
\varsigma e
\chi (\eta ) + \scrI \alpha
\eta e
\chi (\varsigma )
\bigr]
. (3.3)
Proof. Multiplying (3.2) with e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta ), we get
e\varphi (
\varsigma +\eta
2
)e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta ) \leq
\leq e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )e\varphi (\tau \varsigma +(1 - \tau )\eta )+
+e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )e\varphi ((1 - \tau )\varsigma +\tau \eta ).
Integrating with respect to \tau over [0, 1], we have
e\varphi (
\varsigma +\eta
2
)
1\int
0
e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )d\tau \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1222 S. RASHID, M. A. NOOR, K. I. NOOR
\leq
1\int
0
e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )e\varphi (\tau \varsigma +(1 - \tau )\eta )d\tau +
+
1\int
0
e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )e\varphi ((1 - \tau )\varsigma +\tau \eta )d\tau . (3.4)
If we put u = \varsigma \tau + (1 - \tau )\eta , then \tau =
\eta - u
\eta - \varsigma
. So one has
e\varphi (
\varsigma +\eta
2
)
1\int
0
e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )d\tau \leq
\leq 1
\eta - \varsigma
\eta \int
\varsigma
e -
1 - \alpha
\alpha
(\eta - u)e\varphi (u)e\chi (u)du+
1
\eta - \varsigma
\eta \int
\varsigma
e -
1 - \alpha
\alpha
(u - \varsigma )e\varphi (\varsigma +\eta - u)e\chi (u)du =
=
1
\eta - \varsigma
\eta \int
\varsigma
e -
1 - \alpha
\alpha
(\eta - u)e\varphi (u)e\chi (\varsigma +\eta - u)du+
1
\eta - \varsigma
\eta \int
\varsigma
e -
1 - \alpha
\alpha
(u - \varsigma )e\varphi (u)e\chi (u)du =
=
\alpha
\eta - \varsigma
\bigl[
\scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )
\bigr]
.
By the symmetry of the function \chi about
\varsigma + \eta
2
one can see \chi (\varsigma + \eta - x) = \chi (x), x \in [\varsigma , \eta ],
therefore, using this fact and Definition 2.4, we have
e\varphi (
\varsigma +\eta
2
)
\bigl[
\scrI \alpha
\varsigma e
\chi (\eta ) + \scrI \alpha
\eta e
\chi (\varsigma )
\bigr]
\leq \alpha
\eta - \varsigma
\bigl[
\scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )
\bigr]
.
Now multiplying (3.4) with e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta ) and integrating with respect to \tau over [0, 1], we get
1\int
0
e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )e\varphi (\tau \varsigma +(1 - \tau )\eta )d\tau +
+
1\int
0
e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )e\varphi ((1 - \tau )\varsigma +\tau \eta )d\tau \leq
\leq [e\varphi (\varsigma ) + e\varphi (\eta )]
1\int
0
e - \sigma \tau e\chi ((1 - \tau )\varsigma +\tau \eta )d\tau .
From this by setting x = \varsigma (1 - \tau ) + \tau \eta and using e\varphi (\varsigma +\eta - x) = e\varphi (x) it can be seen
\bigl[
\scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )
\bigr]
\leq [e\varphi (\varsigma ) + e\varphi (\eta )]
2
\bigl[
\scrI \alpha
\varsigma e
\chi (\eta ) + \scrI \alpha
\eta e
\chi (\varsigma )
\bigr]
.
Theorem 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
INEQUALITIES INVOLVING NEW FRACTIONAL INTEGRALS TECHNIQUE VIA EXPONENTIALLY . . . 1223
Corollary 3.1. If we choose \alpha = 1, then the inequality (3.3) becomes
e\varphi (
\varsigma +\eta
2
)
\eta \int
\varsigma
e\chi (x)dx \leq
\eta \int
\varsigma
e\varphi (x)e\chi (x)dx \leq e\varphi (\varsigma ) + e\varphi (\eta )
2
\eta \int
\varsigma
e\chi (x)dx.
Remark 3.3. If we take e\chi (x) = 1 and \alpha = 1, then the inequality (3.3) reduces to Theorem 1
(see [6]).
For next result we need the following lemma.
Lemma 3.1. Let \varphi : [\varsigma , \eta ] \rightarrow \BbbR be a differentiable mapping on (\varsigma , \eta ) with \varsigma < \eta . If (e\varphi )\prime \in
\in L1[\varsigma , \eta ], then the following equality for fractional integrals holds:
e\varphi (\varsigma ) + e\varphi (\eta )
2
- (1 - \alpha )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[ \scrI \alpha
\varsigma e
\varphi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )
\bigr]
=
=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) 1\int
0
\bigl[
e - \sigma \tau - e - \sigma (1 - \tau )
\bigr]
e\varphi (\varsigma \tau +(1 - \tau )\eta )\varphi \prime (\varsigma \tau + (1 - \tau )\eta )d\tau . (3.5)
Proof. Consider the right-hand side of (3.5):
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) 1\int
0
\bigl[
e - \sigma \tau - e - \sigma (1 - \tau )
\bigr]
e\varphi (\varsigma \tau +(1 - \tau )\eta )\varphi \prime (\varsigma \tau + (1 - \tau )\eta )d\tau =
=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr)
\left[ 1\int
0
e - \sigma \tau e\varphi (\varsigma \tau +(1 - \tau )\eta )\varphi \prime (a\tau + (1 - \tau )\eta )d\tau -
-
1\int
0
e - \sigma (1 - \tau )e\varphi (\varsigma (1 - \tau )+\tau \eta )\varphi \prime (\varsigma \tau + (1 - \tau )\eta )d\tau
\right] .
Now we compute the first and the second terms of last expression as follows:
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) 1\int
0
e - \sigma \tau e\varphi (\varsigma \tau +(1 - \tau )\eta )\varphi \prime (\varsigma \tau + (1 - \tau )\eta )d\tau =
=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr)
\left[ e - \sigma \tau e\varphi (\tau \varsigma +(1 - \tau )\eta )
\varsigma - \eta
+
\sigma
\varsigma - \eta
1\int
0
e - \sigma \tau e\varphi (\tau \varsigma +(1 - \tau )\eta )d\tau
\right] =
=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr)
\left[ e - \sigma e\varphi (\varsigma ) - e\varphi (\eta )
\varsigma - \eta
+
\sigma
\varsigma - \eta
1\int
0
e - \sigma \tau e\varphi (\tau \varsigma +(1 - \tau )\eta )d\tau
\right] =
=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr)
\left[ e\varphi (\eta ) - e\sigma e\varphi (\varsigma )
\eta - \varsigma
- 1 - \alpha
\alpha
\eta \int
\varsigma
e -
1 - \alpha
\alpha
(\eta - u)e\varphi (u)du
\right] =
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1224 S. RASHID, M. A. NOOR, K. I. NOOR
=
e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \Bigl[ e\varphi (\eta ) - e - \sigma e\varphi (\varsigma ) - (1 - \alpha )\scrI \alpha
\varsigma e
\varphi (\eta )
\Bigr]
.
Analogously we have
\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) 1\int
0
e - \sigma (1 - \tau )e\varphi (\varsigma (1 - \tau )+\tau \eta )\varphi \prime (\varsigma \tau + (1 - \tau )\eta )d\tau =
=
e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \Bigl[ e\varphi (\varsigma ) - e - \sigma e\varphi (\eta ) - (1 - \alpha )\scrI \alpha
\eta e
\varphi (\varsigma )
\Bigr]
.
Hence, the required inequality can be established.
Lemma 3.1 is proved.
Using the above lemma we establish the bounds of a difference of (3.1).
Theorem 3.3. Let \varphi : [\varsigma , \eta ] \rightarrow \BbbR be a differentiable mapping on (\varsigma , \eta ) with \varsigma < \eta such that
e\varphi \in L1[\varsigma , \eta ]. If (e\varphi )\prime is convex on [\varsigma , \eta ], then the following inequality for fractional integral holds:\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| e
\varphi (\varsigma ) + e\varphi (\eta )
2
- (1 - \alpha )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[ \scrI \alpha
\varsigma e
\varphi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )
\bigr] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \biggl\{ \biggl[ (\sigma 2 + 8)e
\sigma
2 + e - \sigma (\sigma 2 + 2\sigma + 4) - (\sigma 2 - 2\sigma + 4)
\sigma 3
\biggr]
\times
\times
\bigl[
| e\varphi (\varsigma )\varphi \prime (\varsigma )| + | e\varphi (\eta )\varphi \prime (\eta )|
\bigr]
+
+
(\sigma 2 - 8)e -
\sigma
2 + 2(\sigma + 2)e - \sigma - 2(\sigma - 2)
\sigma 3
\Theta (\varsigma , \eta )
\biggr\}
,
where
\Theta (\varsigma , \eta ) =
\bigm| \bigm| e\varphi (\varsigma )\varphi \prime (\eta )
\bigm| \bigm| + \bigm| \bigm| e\varphi (\eta )\varphi \prime (\varsigma )
\bigm| \bigm| .
Proof. Using Lemma 3.1 and the exponentially convexity of \varphi , we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| e
\varphi (\varsigma ) + e\varphi (\eta )
2
- (1 - \alpha )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[ \scrI \alpha
\varsigma e
\varphi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )
\bigr] \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq (\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) 1\int
0
| e - \sigma \tau - e - \sigma (1 - \tau )|
\bigm| \bigm| e\varphi (\varsigma \tau +(1 - \tau )\eta )\varphi \prime (\varsigma \tau + (1 - \tau )\eta )
\bigm| \bigm| d\tau \leq
\leq (\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) 1\int
0
| e - \sigma \tau - e - \sigma (1 - \tau )| \times
\times
\bigl\{
\tau | e\varphi (\varsigma )| + (1 - \tau )| e\varphi (\eta )|
\bigr\} \bigl\{
\tau | \varphi \prime (\varsigma )| + (1 - \tau )| \varphi \prime (\eta )|
\bigr\}
d\tau =
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INEQUALITIES INVOLVING NEW FRACTIONAL INTEGRALS TECHNIQUE VIA EXPONENTIALLY . . . 1225
=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) 1\int
0
| e - \sigma \tau - e - \sigma (1 - \tau )|
\Bigl\{
\tau 2| e\varphi (\varsigma )\varphi \prime (\varsigma )| + (1 - \tau 2)| e\varphi (\eta )\varphi \prime (\eta )| +
+\tau (1 - \tau )
\bigl\{
| e\varphi (\eta )\varphi \prime (\varsigma )| + | e\varphi (\varsigma )\varphi \prime (\eta )|
\bigr\} \Bigr\}
d\tau =
=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) 1\int
0
| e - \sigma \tau - e - \sigma (1 - \tau )|
\Bigl\{
\tau 2| e\varphi (\varsigma )\varphi \prime (\varsigma )| +
+(1 - \tau 2)| e\varphi (\eta )\varphi \prime (\eta )| + \tau (1 - \tau )\Theta (\varsigma , \eta )
\Bigr\}
d\tau =
=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr)
\left[
1
2\int
0
[e - \sigma (1 - \tau ) - e - \sigma \tau ]
\Bigl\{
\tau 2| e\varphi (\varsigma )\varphi \prime (\varsigma )| +
+(1 - \tau 2)| e\varphi (\eta )\varphi \prime (\eta )| + \tau (1 - \tau )\Theta (\varsigma , \eta )
\Bigr\}
d\tau +
+
1\int
1
2
\bigl[
e - \sigma \tau - e - \sigma (1 - \tau )
\bigr] \Bigl\{
\tau 2| e\varphi (\varsigma )\varphi \prime (\varsigma )| + (1 - \tau 2)| e\varphi (\eta )\varphi \prime (\eta )| + \tau (1 - \tau )\Theta (\varsigma , \eta )
\Bigr\}
d\tau
\right] :=
:=
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[ K1 +K2
\bigr]
. (3.6)
Calculating K1 and K2, we get
K1 \leq
\bigm| \bigm| e\varphi (\varsigma )\varphi \prime (\varsigma )
\bigm| \bigm|
\left(
1
2\int
0
[e - \sigma (1 - \tau ) - e - \sigma \tau ]\tau 2
\right) d\tau +
+
\bigm| \bigm| e\varphi (\eta )\varphi \prime (\eta )
\bigm| \bigm|
\left(
1
2\int
0
[e - \sigma (1 - \tau ) - e - \sigma \tau ](1 - \tau )2
\right) d\tau +
+ \Theta (\varsigma , \eta )
\left(
1
2\int
0
[e - \sigma (1 - \tau ) - e - \sigma \tau ]\tau (1 - \tau )
\right) d\tau =
=
\bigm| \bigm| e\varphi (\varsigma )\varphi \prime (\varsigma )
\bigm| \bigm| \biggl\{ (\sigma 2 + 8)e -
\sigma
2 - 4(e - \sigma + 1)
2\sigma 3
\biggr\}
+
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1226 S. RASHID, M. A. NOOR, K. I. NOOR
+ | e\varphi (\eta )\varphi \prime (\eta )|
\biggl\{
(\sigma 2 + 8)e -
\sigma
2 - 2e - \sigma (\sigma 2 + 2\sigma + 2) - 2(\sigma 2 - 2\sigma + 2)
2\sigma 3
\biggr\}
+
+ \Theta (\varsigma , \eta )
\biggl\{
(\sigma 2 - 8)e -
\sigma
2 + 2(\sigma + 2)e - \sigma - 2(\sigma - 2)
2\sigma 3
\biggr\}
(3.7)
and
K2 \leq
1\int
1
2
\bigl[
e - \sigma \tau - e - \sigma (1 - \tau )
\bigr] \Bigl\{
\tau 2| e\varphi (\varsigma )\varphi \prime (\varsigma )| + (1 - \tau 2)| e\varphi (\eta )\varphi \prime (\eta )| + \tau (1 - \tau )\Theta (\varsigma , \eta )
\Bigr\}
d\tau =
=
\bigm| \bigm| e\varphi (\varsigma )\varphi \prime (\varsigma )
\bigm| \bigm| \biggl\{ (\sigma 2 + 8)e -
\sigma
2 - 2e - \sigma (\sigma 2 + 2\sigma + 2) - 2(\sigma 2 - 2\sigma + 2)
2\sigma 3
\biggr\}
+
+ | e\varphi (\eta )\varphi \prime (\eta )|
\biggl\{
(\sigma 2 + 8)e -
\sigma
2 - 4(e - \sigma + 1)
2\sigma 3
\biggr\}
+
+ \Theta (\varsigma , \eta )
\biggl\{
(\sigma 2 - 8)e -
\sigma
2 + 2(\sigma + 2)e - \sigma - 2(\sigma - 2)
2\sigma 3
\biggr\}
. (3.8)
Thus, if we use (3.7) and (3.8) in (3.6), we obtain the result.
Theorem 3.3 is proved.
Remark 3.4. For \alpha \rightarrow 1, we find that
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 1
(1 - \alpha )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) =
1
2(\eta - \varsigma )
,
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 1
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \biggl\{ (\sigma 2 + 8)e
\sigma
2 + e - \sigma (\sigma 2 + 2\sigma + 4) - (\sigma 2 - 2\sigma + 4)
\sigma 3
\biggr\}
=
\varsigma - \eta
6
and
\mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow 1
(\eta - \varsigma )e
\sigma
2
4 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \biggl\{ (\sigma 2 - 8)e -
\sigma
2 + 2(\sigma + 2)e - \sigma - 2(\sigma - 2)
\sigma 3
\biggr\}
=
\varsigma - \eta
6
.
Thus, Theorem 3.3 reduces to\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| e
\varphi (\varsigma ) + e\varphi (\eta )
2
- 1
\eta - \varsigma
\eta \int
\varsigma
e\varphi (x)dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq \eta - \varsigma
6
\Bigl[ \bigm| \bigm| e\varphi (\varsigma )\varphi \prime (\varsigma )
\bigm| \bigm| + \bigm| \bigm| e\varphi (\eta )\varphi \prime (\eta )
\bigm| \bigm| +\Theta (\varsigma , \eta )
\Bigr]
.
Our next result about the Pachpatte-type inequality for exponentially convex function for new frac-
tional integral operator technique.
Theorem 3.4. Let \varphi and \chi be real-valued, nonnegative and exponentially convex functions on
[\varsigma , \eta ] \subset \BbbR . Then the following inequality for fractional integral holds:
e\varphi (
\varsigma +\eta
2
)e\chi (
\varsigma +\eta
2
) \leq
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INEQUALITIES INVOLVING NEW FRACTIONAL INTEGRALS TECHNIQUE VIA EXPONENTIALLY . . . 1227
\leq (1 - \alpha )e
\sigma
2
8 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[ \scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )
\bigr]
+
+
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr] [\sigma - 2 + e - \sigma (\sigma + 2)]e
\sigma
2
4\sigma 2 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) +
+
\bigl[
\sigma 2 - 2\sigma + 4 - (\sigma 2 + 2\sigma + 4)e - \sigma
\bigr]
e
\sigma
2
8\sigma 2 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
(3.9)
and
\alpha
\eta - \varsigma
\bigl[
\scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )
\bigr]
\leq
\leq
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr] \sigma 2 - 2\sigma + 4 - e\sigma (\sigma 2 + 2\sigma + 4)
\sigma 3
+
+
e - \sigma (\sigma + 2) + \sigma - 2
\sigma 3
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
. (3.10)
Proof. Using the exponentially convexity of \varphi and \chi on [\varsigma , \eta ], we have
e\varphi (
\varsigma +\eta
2
)e\chi (
\varsigma +\eta
2
) =
= e\varphi (
\tau \varsigma +(1 - \tau )\eta
2
+
(1 - \tau )\varsigma +\tau \eta
2
)e\chi (
\tau \varsigma +(1 - \tau )\eta
2
+
(1 - \tau )\varsigma +\tau \eta
2
) \leq
\leq
\biggl(
e\varphi (\tau \varsigma +(1 - \tau )\eta ) + e\varphi ((1 - \tau )\varsigma +\tau \eta )
2
\biggr) \biggl(
e\chi (\tau \varsigma +(1 - \tau )\eta ) + e\chi ((1 - \tau )\varsigma +\tau \eta )
2
\biggr)
\leq
\leq e\varphi (\tau \varsigma +(1 - \tau )\eta )e\chi (\tau \varsigma +(1 - \tau )\eta )
4
+
e\varphi (\tau \varsigma +(1 - \tau )\eta )e\chi (\tau \varsigma +(1 - \tau )\eta )
4
+
+
\tau (1 - \tau )
2
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr]
+
2\tau 2 - 2\tau + 1
4
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\varsigma )e\chi (\eta )
\bigr]
. (3.11)
Multiplying both sides of (3.11) by e\sigma \tau and then integrating the resulting inequality with respect to
\tau \in [0, 1], we obtain
2 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h} e
\sigma
2
\sigma e
\sigma
2
e\varphi (
\varsigma +\eta
2
)e\chi (
\varsigma +\eta
2
) \leq
\leq
1\int
0
e - \sigma \tau e
\varphi (\tau \varsigma +(1 - \tau )\eta )e\chi (\tau \varsigma +(1 - \tau )\eta )
4
d\tau +
+
1\int
0
e - \sigma \tau e
\varphi (\tau \varsigma +(1 - \tau )\eta )e\chi (\tau \varsigma +(1 - \tau )\eta )
4
d\tau +
+
1\int
0
e - \sigma \tau \tau (1 - \tau )
2
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr]
d\tau +
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1228 S. RASHID, M. A. NOOR, K. I. NOOR
+
1\int
0
e - \sigma \tau 2\tau
2 - 2\tau + 1
4
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\varsigma )e\chi (\eta )
\bigr]
d\tau =
=
\alpha
\eta - \varsigma
\bigl[
\scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )
\bigr]
+
+
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr] \sigma - 2 + e - \sigma (\sigma + 2)
2\sigma 3
+
+
\sigma 2 - 2\sigma + 4 - (\sigma 2 + 2\sigma + 4)e - \sigma
4\sigma 3
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
.
Therefore, we get the inequality
e\varphi (
\varsigma +\eta
2
)e\chi (
\varsigma +\eta
2
) \leq (1 - \alpha )e
\sigma
2
8 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[ \scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )
\bigr]
+
+
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr] [\sigma - 2 + e - \sigma (\sigma + 2)]e
\sigma
2
4\sigma 2 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) +
+
\bigl[
\sigma 2 - 2\sigma + 4 - (\sigma 2 + 2\sigma + 4)e - \sigma
\bigr]
e
\sigma
2
8\sigma 2 \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}
\Bigl( \sigma
2
\Bigr) \bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
,
which completes the proof of (3.9).
Since \varphi and \chi are exponentially convex on [\varsigma , \eta ], then, for \tau \in [0, 1], follows from Definition 2.4
that
e\varphi (\tau \varsigma +(1 - \tau )\eta )e\chi (\tau \varsigma +(1 - \tau )\eta ) \leq
\leq \tau 2e\varphi (\varsigma )e\chi (\varsigma ) + (1 - \tau )2e\varphi (\eta )e\chi (\eta ) + \tau (1 - \tau )
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
and
e\varphi ((1 - \tau )\varsigma +\tau \eta )e\chi ((1 - \tau )\varsigma +\tau b) \leq
\leq (1 - \tau )2e\varphi (\varsigma )e\chi (\varsigma ) + \tau 2e\varphi (\eta )e\chi (\eta ) + \tau (1 - \tau )
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
.
Consequently, we have
e\varphi (\tau \varsigma +(1 - \tau )b)e\chi (\tau \varsigma +(1 - \tau )\eta ) + e\varphi ((1 - \tau )\varsigma +\tau \eta )e\chi ((1 - \tau )\varsigma +\tau \eta ) \leq
\leq (2\tau 2 - 2\tau + 1)
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (b)
\bigr]
+
+2\tau (1 - \tau )
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
. (3.12)
Multiplying both sides of inequality (3.12) by e - \sigma \tau and integrating the resulting inequality with
respect to \tau \in [0, 1], we obtain
1\int
0
e - \sigma \tau e\varphi (\tau \varsigma +(1 - \tau )\eta )e\chi (\tau \varsigma +(1 - \tau )\eta )d\tau +
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INEQUALITIES INVOLVING NEW FRACTIONAL INTEGRALS TECHNIQUE VIA EXPONENTIALLY . . . 1229
+
1\int
0
e - \sigma \tau e\varphi ((1 - \tau )\varsigma +\tau \eta )e\chi ((1 - \tau )\varsigma +\tau \eta )d\tau \leq
\leq
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr] 1\int
0
e - \sigma \tau (2\tau 2 - 2\tau + 1)d\tau +
+
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr] 1\int
0
e - \sigma \tau 2\tau (1 - \tau )d\tau =
=
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr] \sigma 2 - 2\sigma + 4 - e\sigma (\sigma 2 + 2\sigma + 4)
\sigma 3
+
+
e - \sigma (\sigma + 2) + \sigma - 2
\sigma 3
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
.
Thus, we have the inequality
\alpha
\eta - \varsigma
[\scrI \alpha
\varsigma e
\varphi (\eta )e\chi (\eta ) + \scrI \alpha
\eta e
\varphi (\varsigma )e\chi (\varsigma )] \leq
\leq
\bigl[
e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
\bigr] \sigma 2 - 2\sigma + 4 - e\sigma (\sigma 2 + 2\sigma + 4)
\sigma 3
+
+
e - \sigma (\sigma + 2) + \sigma - 2
\sigma 3
\bigl[
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
\bigr]
,
which proves the inequality (3.10).
Theorem 3.4 is proved.
Corollary 3.2. In limiting case, under the assumption of Theorem 3.4, we obtain
2e\varphi (
\varsigma +\eta
2
)e\chi
\bigl(
\varsigma +\eta
2
\bigr)
\leq 1
\eta - \varsigma
\eta \int
\varsigma
e\varphi (x)e\chi (x)dx \leq
\leq e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
6
+
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
3
and
1
\eta - \varsigma
\eta \int
\varsigma
e\varphi (x)e\chi (x)dx \leq e\varphi (\varsigma )e\chi (\varsigma ) + e\varphi (\eta )e\chi (\eta )
3
+
e\varphi (\varsigma )e\chi (\eta ) + e\varphi (\eta )e\chi (\varsigma )
6
.
Conclusion. In this paper, we established the several inequalities within the scope of fractional
integral with exponential kernel. This new fractional integral operator helped in proving the Hermite –
Hadamard type and Hermite – Hadamard – Fejér type inequalities for exponentially convex functions
and in finding bounds of these inequalities. In addition, an immediate consequences of the results
derived in this paper.
Acknowledgment. Authors are pleased to thank the Rector, COMSATS University Islamabad,
Pakistan for providing excellent research and academic environments.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
1230 S. RASHID, M. A. NOOR, K. I. NOOR
References
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Received 27.05.19,
after revision — 23.08.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 9
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| id | umjimathkievua-article-947 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:06Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/b3/e69aa8e1f964d3cc64624a3764a7b9b3.pdf |
| spelling | umjimathkievua-article-9472025-03-31T08:46:40Z Inequalities involving new fractional integrals technique via exponentially convex functions Inequalities involving new fractional integrals technique via exponentially convex functions Rashid, S. Noor, M. A. Noor, K. I. S. Rashid, S. Noor, M. A. Noor, K. I. convex function exponentially convex functions new fractional integral operators Hermite-Hadamard inequality Hermite-Hadamard-Fej´er inequality convex function exponentially convex functions new fractional integral operators Hermite-Hadamard inequality Hermite-Hadamard-Fej´er inequality UDC 517.5 We establish some new Hermite–Hadamard type inequalities involving fractional integral operators with the exponential kernel. Meanwhile, we present many useful estimates on these types of new Hermite–Hadamard type inequalities via exponentially convex functions. &nbsp; УДК 517.5 Нерiвностi, отриманi за допомогою дробових iнтегралiв iз використанням експоненцiально опуклих функцiй За допомогою дробових інтегральних операторів із експоненціальним ядром отримано кілька нерівностей типу Ерміта–Адамара. Серед іншого запропоновано багато корисних оцінок для цих нових нерівностей типу Ерміта–Адамара з використанням експоненціально опуклих функцій. Institute of Mathematics, NAS of Ukraine 2021-09-16 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/947 10.37863/umzh.v73i9.947 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 9 (2021); 1217 - 1230 Український математичний журнал; Том 73 № 9 (2021); 1217 - 1230 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/947/9106 Copyright (c) 2021 saima rashid, Muhammad Aslam Noor, khalida Inayat Noor |
| spellingShingle | Rashid, S. Noor, M. A. Noor, K. I. S. Rashid, S. Noor, M. A. Noor, K. I. Inequalities involving new fractional integrals technique via exponentially convex functions |
| title | Inequalities involving new fractional integrals technique via exponentially convex functions |
| title_alt | Inequalities involving new fractional integrals technique via exponentially convex functions |
| title_full | Inequalities involving new fractional integrals technique via exponentially convex functions |
| title_fullStr | Inequalities involving new fractional integrals technique via exponentially convex functions |
| title_full_unstemmed | Inequalities involving new fractional integrals technique via exponentially convex functions |
| title_short | Inequalities involving new fractional integrals technique via exponentially convex functions |
| title_sort | inequalities involving new fractional integrals technique via exponentially convex functions |
| topic_facet | convex function exponentially convex functions new fractional integral operators Hermite-Hadamard inequality Hermite-Hadamard-Fej´er inequality convex function exponentially convex functions new fractional integral operators Hermite-Hadamard inequality Hermite-Hadamard-Fej´er inequality |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/947 |
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