New fractional inequalities of Hermite–Hadamard type involving the incomplete gamma functions

A specific type of convex functions is discussed. By examining this, we investigate new Hermite–Hadamard type integral inequalities for the Riemann–Liouville fractional operators involving the generalized incomplete gamma functions. Finally, we expose some examples of special functions to support the usefulness and effectiveness of our results.


Introduction
In the past two decades, fractional calculus has received much attention. The fast interest in the topic is due to its extensive applications in various fields such as biochemistry, physics, viscoelasticity, fluid mechanics, computer modeling, and engineering, see [1][2][3][4][5][6][7] for further details. Most of the studies have been devoted to the existence and uniqueness of solutions for fractional differential or difference equations; see e.g. [8][9][10][11][12].
As always, it is important and necessary to specify which model or definition of fractional calculus is being used because there are many different ways of defining fractional operators (integrals and derivatives). To further facilitate the discussion of this model, we present here the definition which is most commonly used for fractional operators, namely the Riemann-Liouville (RL) definition.
In this study, we follow the line of result mentioned above to investigate a new integral inequality, namely, the RL version of the new HH-type inequality (1.8). The rest of the attempt is designed as follows: in Sect. 2.1 we prove the HH inequalities of trapezoidal type by using differintegrals starting from the endpoints of the interval. In Sect. 2.2, we prove the HH inequalities of midpoint type by using differintegrals starting from the midpoint of the interval for the RL-fractional operators. Finally, some applications on special functions are exposed in Sect. 4.

Main results
Our main results are split into two subsections. The following facts will be needed in establishing our main results.

Trapezoidal inequalities
Proof By the exp-convexity of f , we havē Multiplying by ς κ-1 on both sides and then integrating over [0, 1], we get 1 κw Multiplying by κ > 0 on both sides and making the change of variables in the last inequality, we obtain On the other hand, we have by exp-convexitȳ Adding both inequalities, we get Multiplying by ς κ-1 on both sides and then integrating over [0, 1], we get By making the change of variables and Remark 2.1, we get Multiplying by positive constants κ > 0 and (e 1 2 -1) > 0 on both sides, we get Both of inequalities (2.7) and (2.8) rearrange to the required result.
Remark 2.3 The expression (-1) κ γ (κ, -1) occurring in inequality (2.6) may not be clear for the readers, and they will imagine that this value is complex, or does it make sense? Actually, the complex part coming from (-1) κ cancels out the complex part coming from the incomplete gamma γ (κ, -1). Furthermore, this value came from the integral formula (2.1): from looking at the integral we can clearly see that it is real (and positive). Therefore, the answer is yes, it does make sense; the overall expression is real and positive.
On the other hand, we can clarify the above expression by using the Taylor expansion for the integral formula (2.1): .
Thus, our proof is completed.

Midpoint inequalities Proposition Ifw : [ε 3 , ε 4 ] → R is an L 1 and exp-convex function and κ > 0, then we havew
Proof By the exp-convexity of f , we havē Multiplying by ς κ-1 on both sides and then integrating over [0, 1], we get 1 κw Multiplying by κ > 0 on both sides and making the change of variables, we get 2 )-w (ε 3 ) . (2.12) On the other hand, we have by exp-convexitȳ Adding both inequalities, we get Multiplying by ς κ-1 on both sides and then integrating over [0, 1], we get By making the change of variables and Remark 2.2, we get ≤ e 2 κ γ κ, Multiplying by positive constants κ > 0 and (e where j is as before, and δ(κ, j ) = (-1) j j 2 κ+1 γ κ + 1, Proof With the help of Lemma 2.2 and the exp-convexity of |w |, we have Thus, our proof is completed.

He's inequality
This section deals with the HH-inequality in the sense of He's fractional derivatives as introduced in Definition 3.1. As we discussed before, there are many definitions on fractional derivatives in the literature. Herewith we recall the fractional derivatives by the variational iteration method [46,47]. A complete review on variational iteration method and its application and development are available in references [48,49]. Let us recall the following fractional derivative introduced by He [47]. Taking ε 3 = 0 and ε 4 > 0 for all s ∈ (0, 1), multiplying by (ς -s) n-κ-1 (n-κ) on both sides of (3.1), and integrating with respect to t over [0, 1], we get After getting the n -th derivatives on both sides of (3.2) with respect to s and using Definition 3.1, we obtain

Examples
In this section, some examples in the frame of special functions, matrices, and fractional Zakharov-Kuznetsov functions are selected to fulfil the applicability of obtained results. Jρ(z) = 2ρ (ρ + 1)z -v Iρ(z), z ∈ R.