Certain m-Convexity Inequalities Related to Fractional Integrals with Exponential Kernels ()
1. Introduction
Recently, Ahmad et al. [1] presented a new fractional integral operator, which is named fractional integrals with exponential kernels, as follows.
Definition 1.1. Let
. The fractional integrals
and
of order
are defined respectively by
and
Note that
In [1] , the authors obtained new versions of Hermite-Hadanard inequality based on this fractional integral operators as follows.
Theorem 1.1. Let
be a non-negative convex mapping and
. If
, then the following double inequalities hold:
(1.1)
where
.
Taking
i.e.
in Theorem 1.1, we can recapture classical Hermite-Hadamard inequality for a convex function
on
:
(1.2)
This generalized fractional integral operators had attracted the attention of many scholars. For example, Wu et al. [2] gave three fundamental integral identities via fractional integrals with exponential kernels to establish several Hermite-Hadamard-type inequalities. Zhou et al. [3] derived some parameterized fractional integrals with exponential kernels inequalities for convex mappings. For more information associated with fractional integrals with exponential kernels see reference in [4] [5] .
The concept of m-convex mappings was introduced by Toader in [6] . It is defined as follows.
Definition 1.2. The mapping
,
is named m-convex mapping, where
, if for all
and
, we have
(1.3)
Due to the wide applications of m-convex mapping, many authors have established various integral inequalities related to m-convex mappings. In [7] , Dragomir presented some properties and inequalities for m-convex mappings. In [8] , Jleli et al. extended partial results presented in [7] via generalized fractional integrals. In [9] , Farid and Abbas gave some general fractional integral inequalities for m-convex mappings associated with generalized Mittag-Leffer mapping. For other works involving m-convex mappings, we refer an interseted reader to [10] [11] [12] .
These studies motivated us to establish some fractional integrals with exponential kernels inequalities for m-convex mappings. We considered two forms of m-convex combination to get certain midpoint type and trapezoid type inequalities. We gave new bounds for these inequalities and laid a foundation for their application in numerical integration. Some results of this article would provide generalizations of those given in earlier works.
2. Main Results
In this part, we mainly establish some fractional integral inequalities based on the properties of m-convex functions.
Theorem 2.1. Let
be a m-convex function with
and
. If
, then the following inequality exists:
(2.1)
where
.
Proof. By means of m-convexity of
, one has
and
Adding the above inequalities, we deduce
(2.2)
We can obtain the desired inequality by multiplying (2.2) with
and then integrating over
with respect to
. Since
and
This ends the proof.
Corollary 2.1. If we consider
in Theorem 2.1, then we have right part of inequality (8) in [1] .
To obtain trapezoid type inequality related to fractional integrals with exponential kernels, we need the following lemma.
Lemma 2.1. Assuming
is a differentiable mapping with
and
. If
, then the following identity holds:
(2.3)
where
and
(2.4)
Proof. Integrating the following formula by parts, we have
(2.5)
Multiplying both sides of (2.5) by
, we have the conclusion (2.3). The proof is completed.
Theorem 2.2. Let
be defined as in Lemma 2.1. If
is m-convex on
for some fixed
, then the following inequality for fractional integrals with exponential kernels holds:
(2.6)
Proof. Applying Lemma 2.1 and convexity of
, we obtain
(2.7)
By calculation, we have
(2.8)
and
(2.9)
Utilizing inequality (2.8) and inequality (2.9) in inequality (2.7), we have
The proof is completed.
Corollary 2.2. If we consider
in Theorem 2.2, then we can deduce Theorem 3 in [1] .
Theorem 2.3. Let
be a m-convex function with
and
. If
, then the following inequality exists:
(2.10)
where
.
Proof. By means of m-convexity of
, we deduce
and
Multiplying above-mentioned inequalities with
and then integrating over
with respect to
, we get
(2.11)
and
(2.12)
By adding (2.11) and (2.12) together, we have completed the proof.
Corollary 2.3. If we consider
i.e.
in Theorem 2.3, observe that
and
then we have
(2.13)
Theorem 2.4. Under the assumptions of Theorem 2.3, if we take
, then the resulting expression holds:
(2.14)
Proof. Since
is m-convex, we have
(2.15)
and
(2.16)
Adding inequality (2.15) and inequality (2.16) together and then multiplying by
, we get
(2.17)
Integrating on both sides of inequality (2.17) respect to
over
, we have completed the proof. Since
and
We now use the following two lemmas, which are presented in [13] , to obtain some mid-point type and trapezoid type inequalities.
Lemma 2.2. Assuming
is a differentiable mapping, such that
with
, then the following identity holds:
(2.18)
Lemma 2.3. Assuming
is a positive convex mapping, such that
with
, then the following identity exists:
(2.19)
Theorem 2.5. Under the assumptions of lemma 2.2, if
is m-convex on
, then the resulting expression holds:
(2.20)
Proof. Applying Lemma 2.2 and the convexity of
, one has
where we use the fact that
and
After suitable arrangements, we obtain
This ends the proof.
Theorem 2.6. Under the assumptions of Lemma 2.3, if
is m-convex on
, then the resulting expression holds:
(2.21)
Proof. Applying Lemma 2.3 and the convexity of
, one has
where we use the fact that
and
After suitable arrangements, we obtain
This ends the proof.
3. Conclusion
In this article, taking different exponential kernels parameters, we established three fractional integrals inequalities for m-convex mappings. Furthermore, we constructed a new lemma to obtain Dragomir-Agarwal inequality for m-convex mappings. We emphasized that certain results proved in this article generalize and extend parts of the results provided by Ahmad et al. in [1] . Finally, we gave mid-point type and trapezoid type inequalities for m-convex mappings.
Funding
This work is supported by the General project of Education Department of Hunan Province (No. 19C0359) and General project of Education Department of Hunan Province (No. 19C0377).
NOTES
*Corresponding author.