On the Generalized Hermite–Hadamard Inequalities via the Tempered Fractional Integrals

: Integral inequality plays a critical role in both theoretical and applied mathematics ﬁelds. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the ﬁeld of inequalities due to the behaviour of its deﬁnition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we ﬁrst introduced the notion of λ -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite–Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann–Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modiﬁed Bessel functions and q -digamma function.


Introduction
Leth : J ⊆ R → R be a convex function and 3 , 4 ∈ J with 3 < 4 . Then, the well known inequalities, namely, the Hermite-Hadamard inequalities [1], defined bȳ We recall that the Hermite-Hadamard inequalities are related to the integral mean of a convex function. This provides an estimate from both sides of the mean value and assures the integrability of a convex function. Several classical inequalities can be obtained with the help of Hadamard's inequality considering the use of peculiar convex functionsh. Moreover, these inequalities for convex functions have a very important role in both applied and pure mathematics. Typical applications of the classical inequalities are: probabilistic problems, decision making in structural engineering and fatigue life.
The right and left part inequality of the inequalities (1) are called trapezoidal and midpoint inequalities. Researchers have ben working on two types of inequalities (1). Many of them have been worked only on the trapezoidal type inequality [2][3][4] or the midpoint type inequality [5,6] while the others have been workingd on both of them at the same time [7][8][9]. Both trapezoidal and midpoint inequalities can be explained using the following definition Definition 1 ([7]). Supposeh : [ 3 , 4 ] ⊂ R → R is a twice differentiable function on an open interval ( 3 , 4 ) with the second derivative bounded on the interval ( 3 , 4 ); that is, h ∞ := sup x∈( 3 , 4 ) h (x) < ∞, then, the trapezoidal and midpoint type inequalities are defined by respectively.
From a complementary viewpoint to Ostrowski type inequalities [10], trapezoidal and midpoint type inequalities provides a priory error bounds in estimating the Riemann integralby a generalized midpoint and trapezoidal formula [7,11]. We know that the development of Ostrowski's inequality has experienced attractive growth in the past decade, with over two thousands papers on it. A large number of refinements, generalizations, and extensions in both discrete and integral cases have been discovered (see [8,9]). Generalised versions have been discussed, e.g., the corresponding versions on time scales, form-time differentiable functions, for multiple integrals or vector valued functions as well (see [7,12]). Numerous applications in numerical analysis, special functions, probability theory, and other fields have been also given (see [8]).
First, let us recall the above definition of the Riemann-Liouville fractional integrals (left and right) which are defined by [15,17]: where the gamma function is defined as Now, let us recall the basic expressions of Hermite-Hadamard inequality for fractional integrals is proved by Sarikaya et al. in [3] as follows.
Theorem 3. Ifh : [ 3 , 4 ] → R is an L 1 function with 3 < 4 . Then, we havē Meanwhile, in [3], Sarikaya et al. established the following trapezoidal type equality and inequality for Riemann-Liouville integral, respectively: On the other hand, the equality and inequalities of midpoint type are pointed out in Remarks 10-12. Now, we recall the basic definitions and new notations of tempered fractional operators. Definition 2 ([35,36]). Let [ 3 , 4 ] be a real interval and λ ≥ 0, ν > 0. Then for a functionh ∈ L 1 [ 3 , 4 ], the left and right tempered fractional integral, respectively, defined by We recall that several researchers the Riemann-Liouville fractionals integrals and provided important generalizations of Hermite-Hadamard type inequalities utilising these type of integrals for various type of convex functions, see, for instance, [3,19,20]. There is a strong relationship between convexity and symmetry. Which ever one we work on we can apply to the other due to the strong correlation produced between them especially in recent years (see [37]).
In this article, we followed the Sarikaya et al. [3] and Sarikaya and Yildirim [6] technique to establish a few inequalities of Hermite-Hadamard type (including both trapezoidal and midpoint type) which involved the tempered fractional integrals and the notion of λ-incomplete gamma function for convex functions. During the research, we found that our findings generalise the previous findings in the literature and this fact can be observed in Remarks 2-12.
The rest of this article is designed as: In Section 2.1, we obtain the inequalities of trapezoidal-and midpoint-type using integrals starting from the endpoints of the given interval, and in Section 2.2, we obtain analogous results using integrals starting from the midpoint of the given interval and some other relevant findings. Section 3, includes the application of our obtained results in special functions. The discussion on the proposed findings and concluding remarks are given in Section 4.

Hermite-Hadamard Inequalities Involving Beta Function
First of all we define the new incomplete gamma function: Definition 3. For the real numbers ν > 0 and x, λ ≥ 0, we define the λ-Incomplete gamma function by If λ = 1, it reduces to the incomplete gamma function [38]:
(i) The first item's proof follows from the Definition 3 and changing the variable u := ( 4 − 3 )χ.
(ii) From the Definition 3, we have By changing the order of the integration, we get Making the use of Remark 1 (i) we get This ends the proof of the second item.
In the next two subsections, we obtain some integral inequalities involved the λ-Incomplete gamma function.

Inequalities of
4 -Type In this section, we prove a few inequalities of trapezoidal type or for ν > 0, λ ≥ 0.
Proof. The convexity ofh allows us to writē Multiplying both sides of (13) byχ ν−1 e −λ( 4 − 3 )χ , and then integrating both sided with respect toχ over [0, 1] to get As consequence, we obtain and so we have proved the first part inequality.

Remark 5. Identity
In the extensions of our results, we have: where L (ν,λ) 1 Proof. Making the use of Lemma 4 and the convexity of |h |, we deduce where we have used the inequality in Appendix A. Thus our proof is done. where Proof. By making the use of Lemma 4, Hölder's inequality and the convexity of |h | , we deduce which completes the proof of (17). (17) becomes the inequality (3) for λ = 0 and ν = 1. (17)
On the other hand, we have from the convexity ofh that Adding these to get We multiply both sides byχ ν−1 e −λ( 4 − 3 )χ and then integrating with respect toχ overχ ∈ [0, 1] to deduce Again, by making the use of u :=χ 2 3 + 2−χ 2 4 , v := 2−χ 2 3 +χ 2 4 and Remark 1 (ii) in the last inequality, we get and this ends the proof of the second inequality. Therefore, the inequalities (18) is proved.
Proof. By making the use of integrating by parts and Remark 1 (i), we get Analogously, we have Consequently, we have This ends the complete proof of Lemma 5.
Remark 11. If in Lemma 5, we set 1. λ = 0, then equality (19) becomes the following equality 2. λ = 0 and ν = 1, then equality (19) becomes the following equality these are both done by Sarikaya and Yildirim in [6]. where Proof. At first, we let = 1. Then by making the use of Lemma 5, Remark 1 (ii) and the convexity of |h | , we have For > 1, we use the Lemma 5, power-mean inequality, Remark 1 (ii) and the convexity of |h | to get where we have used the identities in Appendix B. Thus our proof is done.

Examples
There are many applications to demonstrate the use of integral inequalities, especially applications on special means of the real numbers [2,5,8,39]. In this section, we present some examples to demonstrate the applications of our proposed results on modified Bessel functions and q-digamma functions.
for 0 < < 1, and From this definitions, we see that z → Ψ (z) is a completely monotonic function on an interval (0, ∞) for all > 0, and consequently, z → Ψ (z) is convex on the same interval.

Conclusions
In this article, we introduced an extension of the well known incomplete gamma function, namely the λ-incomplete gamma function to connect with the model of tempered fractional integrals. In view of this, we considered the integral inequalities of Hermite-Hadamard type in the context of tempered fractional integrals. Integral inequalities form a crucial branch of analysis and were combined with various types of fractional integrals but we had never seen this before with tempered fractional integrals. For this reason, we studied the inequality of Hermite-Hadamard type and related inequalities via the tempered fractional integrals which generalized the previous results obtained in [2,3,5,6].

Appendix A
By using the properties of modulus and the convexity of |h |, we can deduce:

Appendix B
By changing the order of the integration (just like Remark 1 (ii)), we have