Certain mean-type fractional integral inequalities via different convexities with applications

In this paper, we establish certain generalized fractional integral inequalities of mean and trapezoid type for (s + 1)-convex functions involving the (k, s)-Riemann–Liouville integrals. Moreover, we develop such integral inequalities for h-convex functions involving the k-conformable fractional integrals. The legitimacy of the derived results is demonstrated by plotting graphs. As applications of the derived inequalities, we obtain the classical Hermite–Hadamard and trapezoid inequalities.


Introduction
The well known Hermite-Hadamard inequality for a convex function Ψ : U → R on an interval U of real numbers, with φ, ϕ ∈ U and φ < ϕ is given by (1.1) Numerous scientists examined this inequality and published various generalizations and extensions by using fractional integrals and derivatives [5, 8, 15, 16, 18, 19, 23, 25-29, 32, 33]. The theories of k-and (k, s)-fractional operators are the more generalized way to express fractional calculus operators (see [21,22,24]). The classical fractional operators become special cases of such theories. Considering late developments in the theory of differential and integral equations, it is getting very hard to ignore the existence of integral inequalities that help determine the bounds on unknown functions. Applications of integral inequalities are important in various fields of science, like mathematics, physics, engineering, among others, we especially notice initial-value problems, the stability of linear transformation, integral differential equations, and impulse equations. We refer the readers [1,3,4,6,7,12,13,20] for such applications in several branches of mathematics and the references therein. Firstly, we give some key definitions and mathematical fundamentals of the theory of fractional calculus which are utilized in this paper.
Sanja Varošanec presented the class of convex functions in [31] as follows: The formal definition of the beta function given in [2] is stated as follows: The classical beta function, also called the Euler integral of the first kind, is a special function defined by The integral form of the hypergeometric function is given as for all φ, ϕ ∈ J and ℘ ∈ [0, 1].
(ii) The proof of (ii) is similar to (i), so is omitted. Thus, the proof of the theorem is completed.
Example 2.2 By plotting the graphs of (2.1) for a convex function ψ(℘) = e ℘ , we check that both inequalities are valid. It is known that the (k, s)-Riemann-Liouville fractional integrals of this function for s = 0 are given by (2.9) By utilizing these expressions in the double inequality (2.1), we get (2.10) The three functions given by the left, middle, and right sides of the double inequality (2.10) are plotted in Fig. 1 against χ ∈ (0, 1]. The graphs of the functions show the validity of dual inequality.

Theorem 2.4 Let
(2.14) (ii) for s < -1, Proof (i) Applying Theorem 2.3, modulus property, Hölder's inequality, and (s + 1)convexity of |ψ | g , we get This completes the proof of (i). (ii) The proof of (ii) is similar to (i), so is omitted. Thus, the proof of the theorem is completed.
Example 2.5 By plotting the graphs of inequalities of Theorem 2.4 for the convex function ψ(℘) = ℘ 2 and g = 2, we prove the validity of the results. Substitution of (2.8) and (2.9) into inequality (2.14) gives The three functions given by the left, middle and right sides of the double inequality (2.15) are plotted in Fig. 2 against χ ∈ (0, 1]. The graphs of the functions illustrate the validity of both inequalities.

Inequalities involving conformable fractional integral operator
This section contains mean-type inequalities for conformable fractional integral operators by using h-convexity.
Proof Since ψ is h-convex function, we can write and .
Example 3.2 We verify the result of Theorem 3.1 for the convex function ψ(℘) = e 2℘ and h(ν) = ν. It is known that the conformable fractional integrals of this function for β = 1 are given by and Substituting these expressions into inequality (3.1), we get (3.10) The three functions given by the left, middle, and right sides of this double inequality are plotted in Fig. 3 against χ ∈ (0, 1] to show clearly that both inequalities are valid. If |ψ | is an h-convex function on [φ, ϕ], then the following inequality for fractional con- Figure 3 The graphs illustrate the validity of the dual inequality (3.10) for the case φ = 0 and ϕ = 1 formable integrals holds: Proof By using Lemma 1.10, modulus property, and h-convexity, we have which completes the proof.
Example 3. 4 We verify the result of Theorem 3.3 for the convex function ψ(ν) = e ν , β = 1, and h(ν) = ν. In this case, inequality (3.11) is given by The three functions given by the left, middle, and right sides of this double inequality are plotted in Fig. 4 against χ ∈ (0, 1] to show that both inequalities are valid.

Figure 4
The graphs illustrate validity of the double inequality given by (3.12) for the case φ = 0, ϕ = 1, and 0 < χ ≤ 1 Corollary 3.5 If we take h(℘) = 1 in Theorem 3.3, then we get the result: where B denotes the usual beta function.

Corollary 3.6
If we take h(℘) = ℘ in Theorem 3.3, then we get the result for simple convex function presented below:

Inequalities involving generalized k-conformable fractional integral operators
In this section, mean-type inequalities for k-conformable fractional integral operator by using h-convexity are established.
Proof Since ψ is an h-convex function, we can write  Integrating by parts and using substitution u = ℘φ + (1 -℘)ϕ, we have Also, we have By using the obtained values of I 1 and I 2 in (4.7) and then multiplying the result by ϕ-φ 2 , we get the desired result. If |ψ | is an h-convex function on [φ, ϕ], then the following inequality for fractional con-findings of this investigation complement those of previous studies. Simply, the recent study confirms the earlier results and plays an additional role by making generalizations.