Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.


Introduction
The inequality discovered by Hermite and Hadamard (see [3], [4, pp. 137]) is one of the best-established inequalities in the theory of convex functions with a geometrical interpretation and many applications. These inequalities state that, if f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b, then (1.1) interval enclosure is Archimedes' method which is related to computation of the circumference of a circle. In 1966, the first book related to interval analysis was given by Moore who is known as the first user of intervals in computational mathematics; see [15]. After his book, several scientists started to investigate the theory and application of interval arithmetic. Nowadays, because of its applications, interval analysis is a useful tool in the various area which are interested intensely in uncertain data. You can see applications in computer graphics, experimental and computational physics, error analysis, robotics, and many others. Moreover, several important inequalities (Hermite-Hadamard, Ostrowski, etc.) have been studied for the interval-valued functions in recent years. In [16,17], Chalco-Cano et al. obtained Ostrowski type inequalities for interval-valued functions by using the Hukuhara derivative for interval-valued functions. In [18], Román-Flores et al. established Minkowski and Beckenbach's inequalities for interval-valued functions. For the others, see [18][19][20][21][22]. However, inequalities were studied for the more general set-valued maps. For example, in [23], Sadowska gave the Hermite-Hadamard inequality. For other studies, you can see [24,25].
The purpose of this paper is to complete the Riemann-Liouville integrals for intervalvalued functions and to obtain the Hermite-Hadamard inequality via these integrals. Furthermore, Hermite-Hadamard-type inequalities are given using these integrals.

Interval calculus
A real valued interval X is a bounded, closed subset of R and is defined by where X, X ∈ R and X ≤ X. The numbers X and X are called the left and the right endpoints of the interval X, respectively. When X = X = a, the interval X is said to be degenerate and we use the form X = a = [a, a]. Also, we call X positive if X > 0 or negative if X < 0. The set of all closed intervals of R, and the sets of all closed positive intervals of R and closed negative intervals of R is denoted by R I , R + I and R -I , respectively. The Pompeiu-Hausdorff distance between the intervals X and Y is defined by It is well known that (R I , d) is a complete metric space; see [26]. Now, we give the definitions of basic interval arithmetic operations for the intervals X and Y as follows: where λ = -1. The subtraction is given by In general, -X is not additive inverse for X, i.e. X -X = 0. The definitions of operations lead to a number of algebraic properties which allows R I to be a quasilinear space; see [27]. They can be listed as follows (see [15,[26][27][28][29]): (1) (Associativity of addition) (X + Y ) + Z = X + (Y + Z) for all X, Y , Z ∈ R I , (2) (Additivity element) X + 0 = 0 + X = 0 for all X ∈ R I , (3) (Commutativity of addition) X + Y = Y + X for all X, Y ∈ R I , (4) (Cancelation law) X + Z = Y + Z ⇒ X = Y for all X, Y , Z ∈ R I , (5) (Associativity of multiplication) (X · Y ) · Z = X · (Y · Z) for all X, Y , Z ∈ R I , (6) (Commutativity of multiplication) X · Y = Y · X for all X, Y ∈ R I , (7) (Unity element) X · 1 = 1 · X for all X ∈ R I , (8) (Associativity law) λ(μX) = (λμ)X for all X ∈ R I and all λ, μ ∈ R, (9) (First distributivity law) λ(X + Y ) = λX + λY for all X, Y ∈ R I and all λ ∈ R, (10) (Second distributivity law) (λ + μ)X = λX + μX for all X ∈ R I and all λ, μ ∈ R. Besides these properties, the distributive law is not always valid for intervals. For example, X = [1,2], Y = [2,3] and Z = [-2, -1]. 4] whereas X · Y + X · Z = [- 2,5].
But this law holds in certain cases. If Y · Z > 0, then Moreover, one of the set properties is the inclusion ⊆ that is given by Considering together with arithmetic operations and inclusion, one has the following property which is called the inclusion isotone of interval operations: Let be the addition, multiplication, subtraction or division. If X, Y , Z and T are intervals such that X ⊆ Y and Z ⊆ T, then the following relation is valid: The following proposition is about scalar multiplication preserving the inclusion. Proposition 1 Let X, Y be intervals and λ ∈ R. If X ⊆ Y , then λX ⊆ λY .

Integral of interval-valued functions
In this section, the notion of integral is mentioned for interval-valued functions. Before the definition of integral, the necessary concepts will be given as follows: A function F is said to be an interval-valued function of t on [a, b], if it assigns a nonempty interval to each t ∈ [a, b] A partition of [a, b] is any finite ordered subset P having the form P : a = t 0 < t 1 < · · · < t n = b.
We denote by P( [a, b]) the set of all partitions of [a, b]. Let P(δ, [a, b]) be the set of all P ∈ P([a, b]) such that mesh(P) < δ. Choose an arbitrary point ξ i in the interval [t i-1 , t i ], (i = 1, 2, . . . , n) and let us define the sum where F : [a, b] → R I . We call S(F, P, δ) a Riemann sum of F corresponding to P ∈ P(δ, [a, b] The collection of all functions that are (IR)-integrable on [a, b] will be denoted by IR ([a,b]) .
Otherwise, Zhao et al. obtained the following Hermite-Hadamard inequality for interval-valued functions by using h-convexity.
2) reduces to the following result: which is obtained by Sadowska in [23].
(ii) If h(t) = t s , then (2.2) reduces to the following result: which is obtained by Osuna-Gomez et al. in [36].
Remark 2 If h(t) = t, then (2.5) reduces to the following result: Remark 3 If h(t) = t, then (2.4) reduces to the following result: (2.8) , then for α > 0 we have where M(a, b) and N(a, b) are defined in Theorem 3.
For the other fractional inequalities for the convex interval-valued functions, see [37].

Interval-valued approximately convexities
In this section we define a new class of interval-valued approximately h-convex functions, which is depending upon a given function. We discuss some special cases of our new definition and find new definitions of approximately h-convex functions. We let (X, · I ) be a normed quasilinear space, let I be a nonempty interval-valued convex subset of X, H : X × X → R and let h : (0, 1) → R be the given functions.

Definition 3 A function F
for all t ∈ (0, 1) and a, b ∈ I. Now we discuss some special cases of Definition 3. I. If we use H(x, y) = ( xy ) γ for some ∈ R and γ > 1 in Definition 3, then we have a new definition of an interval-valued approximately convex function which is called an interval-valued γ -approximately h-convex function.
Definition 4 A function F : I → R + I is said to be an interval-valued γ -approximately hconvex function, if for all t ∈ (0, 1) and a, b ∈ I.
II. If we use H(x, y) = ( xy ) for some ∈ R in Definition 3, then we have a new definition of an interval-valued approximately convex function which is called an interval-valued -approximately h-convex function.
III. If we use H(x, y) = -μt(1t) yx 2 for some μ > 0 in Definition 3, then we have a new definition of an interval-valued approximately convex function which is called an interval-valued strongly h-convex function.

Definition 6 A function
for all t ∈ (0, 1) and a, b ∈ I.
IV. If we use H(x, y) = μt(1t) yx 2 for some μ > 0 in Definition 3, then we have a new definition of interval-valued approximately convex function which is called an interval-valued relaxed h-convex function.
Definition 7 A function F : I → R + I is said to be an interval-valued relaxed h-convex function, if for all t ∈ (0, 1) and a, b ∈ I.
V. If we use γ = 0 in Definition 4 or = 0 in Definition 5, we have Definition 2.

Interval-valued generalized fractional integral operators
In this section, we define a generalized fractional integral operator for the interval-valued functions and discuss special cases of our newly define integral operator. Let us define a function ϕ : [0, +∞) → [0, +∞) satisfying the following conditions: where A 1 , A 2 , A 3 > 0 are independent of r, s > 0. If ϕ(r)r α is increasing for some α ≥ 0 and ϕ(r) r β is decreasing for some β ≥ 0, then ϕ satisfies (4.1)-(4.4); see [12]. Meanwhile, in [11], Sarikaya and Ertuğral defined the following generalized fractional integrals: Therefore, we can give the following new definitions.
and F ∈ IR ([a,b]) . Then the interval-valued left-sided and right-sided generalized fractional integrals of the function F, respectively, are given as . Then, we have The most important feature of interval-valued generalized fractional integrals is that they generalize some types of fractional integrals such as the Riemann-Liouville fractional integral, the k-Riemann-Liouville fractional integral, Katugampola fractional integrals, the conformable fractional integral, and Hadamard fractional integrals. These important special cases of the integral operators (4.5) and (4.6) are mentioned below.

Main results
In this section, we prove some inequalities of Hermite-Hadamard type for the intervalvalued approximately h-convex functions via generalized fractional integrals. We use for brevity the following notations in the next new results: , then we have the following inequalities for the generalized fractional integrals: H(x, y).
By setting x = ta + (1t)b and y = tb + (1t)a in (5.2), we obtain Multiplying both sides of (5 and integrating the resultant one with respect to t over [0, 1], we get In Eq. (5.4), using Theorem 1, we obtain Similarly, we have Hence, we achieved our first inequality. To prove the second inequality since F is intervalvalued approximately h-convex function, we get and H(a, b).
Adding (5.5) and (5.6), we have on both sides and integrating the resultant one with respect to t over [0, 1], we have This completes the proof.

Corollary 2
If we choose ϕ(t) = t in Theorem 6, then we have the following inequalities: H(a, b).
in Theorem 6, then we have the following inequalities for the Riemann-Liouville fractional integrals: , then we have the following inequality for the generalized fractional integrals: where M(a, b) and N(a, b) are defined in Theorem 3 and H(a, b).
on both sides and integrating the resultant one with respect to t over [0, 1], we get Using Theorem 1 in Eq. (5.17), we have .
in Theorem 7, then we have the following inequality for the Riemann-Liouville fractional integrals: , then we have the following inequality for the generalized fractional integrals: N(a, b) and K 1 , K 2 , K 3 , P(a, b) are defined from Theorem 3 and Theorem 7, respectively.
Proof For t ∈ [0, 1], we can write Since F and G are two interval-valued approximately h-convex functions, we have

Multiplying by ϕ((b-a)t)
t both sides of inequality (5.23) and integrating the resultant one with respect to t over [0, 1], we obtain By changing the variable of integration we achieved the desired inequality (5.22).

Corollary 6
Under the assumptions of Theorem 8 with ϕ(t) = t, then we have the following inequality: Corollary 7 Under the assumptions of Theorem 8 with ϕ(t) = t α (α) , then we have the following inequality for the Riemann-Liouville fractional integrals: , then we have the following inequalities for the generalized fractional integrals: y).
Multiplying by both sides of inequality (5.27) and integrating the resultant one with respect to t over [0, 1], we obtain Using Theorem 1 and Eq. (5.28), we have Similarly, we get Hence, we proved the first inequality. To prove the second inequality of (5.26), first we note that, since F is an interval-valued approximately h-convex function, we have H(a, b).
Adding (5.29) and (5.30), we get Multiplying by both sides of inequality (5.31) and integrating the resultant one with respect to t over [0, 1], we obtain By changing the variables of integration we have the second inequality of (5.26).

Corollary 8
If we choose ϕ(t) = t in Theorem 9, then we have the following inequalities: H(a, b).

Corollary 9
Taking ϕ(t) = t α (α) in Theorem 9, then we have the following inequalities for the interval-valued fractional operators: H(a, b).

Theorem 10 If F, G : [a, b] → R + I are two interval-valued approximately h-convex functions such that F(t) = [F(t), F(t)] and G(t) = [G(t), G(t)], then we have the following inequality for the generalized fractional integrals:
where M(a, b) and N(a, b) are defined in Theorem 3 and Proof Since F and G are two interval-valued approximately h-convex functions, H(a, b). (5.36) Multiplying (5.35) and (5.36), we have H 2 (a, b).

Multiplying by
both sides of inequality (5.39) and integrating the resultant one with respect to t over [0, 1], we have By using Theorem 1 in Eq. (5.40), we obtain our required inequality.

Corollary 10
Taking ϕ(t) = t in Theorem 10, then we have the following inequality:

Corollary 11
Taking ϕ(t) = t α (α) in Theorem 10, then we have the following inequality for the interval-valued fractional operators: M(a, b), N(a, b)

Theorem 11 If F, G : [a, b] → R + I are two interval-valued approximately h-convex functions such that F(t) = [F(t), F(t)] and G(t) = [G(t), G(t)], then we have the following inequality for the generalized fractional integrals:
H(x, y).

Multiplying by
both sides of inequality (5.47) and integrating the resultant one with respect to t over [0, 1], we obtain our result (5.43).

Corollary 12
Taking ϕ(t) = t in Theorem 11, then we have the following inequality: a, b).

Corollary 13
Taking ϕ(t) = t α (α) in Theorem 11, then we have the following inequality for the interval-valued fractional operators:

Theorem 12 If F : [a, b] → R + I is interval-valued approximately h-convex function such that F(t) = [F(t), F(t)], then we have the following inequalities for the generalized fractional
integrals: H(a, b).
Proof Since F is an interval-valued approximately h-convex function on [a, b], we have H(x, y).
For x = 1-t 2 a + 1+t 2 b and y = 1+t 2 a + 1-t 2 b, we get Multiplying by both sides of inequality (5.51) and integrating the resultant one with respect to t over [0, 1], we obtain By using Theorem 1 in Eq. (5.52), we have Similarly, we get Hence, we proved the first inequality. To prove the second inequality of (5.50), first we note that, since F is an interval-valued approximately h-convex function, we have H(a, b).

Multiplying by
both sides of inequality (5.55) and integrating the resultant one with respect to t over [0, 1], we obtain This completes the proof.

Corollary 14
If we choose ϕ(t) = t in Theorem 12, then we have the following inequalities: H(a, b).

Corollary 15
Taking ϕ(t) = t α (α) in Theorem 12, then we have the following inequalities for the interval-valued fractional operators: H(a, b).
where M(a, b) and N(a, b) are defined in Theorem 3 and Proof Since F and G are two interval-valued approximately h-convex functions, H(a, b).
Adding (5.61) and (5.62), we obtain the following relation: Multiplying by both sides of inequality (5.63) and integrating the resultant one with respect to t over [0, 1], we have By using Theorem 1 in Eq. (5.64), we obtain our required inequality.

Corollary 16
If we choose ϕ(t) = t in Theorem 13, then we get the following inequality: a, b).

two interval-valued approximately h-convex functions such that F(t) = [F(t), F(t)] and G(t) = [G(t), G(t)], then we have the following inequality for the generalized fractional integrals:
where M(a, b), N(a, b) and K 7 , K 8 , K 9 are defined in Theorem 3 and Theorem 13, respectively.
Proof Since F and G are two interval-valued approximately h-convex functions on [a, b], we have H(x, y). (5.68) For x = 1-t 2 a + 1+t 2 b and y = 1+t 2 a + 1-t 2 b, we obtain Similarly, we get Multiplying the inequalities (5.69) and (5.70), we obtain H(a, b) Multiplying by both sides of inequality (5.71) and integrating the resultant one with respect to t over [0, 1], we obtain our result (5.67).

Corollary 18
Taking ϕ(t) = t in Theorem 14, then we have the following inequality:

Corollary 19
Taking ϕ(t) = t α (α) in Theorem 14, then we have the following inequality for the interval-valued fractional operators:

Some special cases
In this section, we discuss some special cases from our main results. From Theorem 6, we have the following result.
From Theorem 8, we obtain the following result.
From Theorem 9, we have the following result.

Corollary 26
If F : [a, b] → R + I is interval-valued -approximately h-convex function, then From Theorem 10, we get From Theorem 11, we obtain the following result.

Corollary 32 If F : [a, b] → R + I is interval-valued -approximately h-convex function, then
From Theorem 13, we get the following result.