In the present paper we initiate the study of boundary value problems like (.)-(.), in which we combine Riemann-Liouville **fractional** diﬀerential equations subject to the **Hadamard** **fractional** **integral** boundary conditions. The key tool for this combination is Property . from [], p.. To the best of the authors’ knowledge this is the ﬁrst paper dealing with the Riemann-Liouville **fractional** diﬀerential equation subject to **Hadamard** type **integral** boundary conditions.

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Theorem 3.1 Let f and h be two positive continuous functions on the interval [1, ∞) and f ≤ h on [1, ∞). If h f is decreasing and f is increasing on [1, ∞), then for a convex function Φ with Φ(0) = 0, the generalized proportional **Hadamard** **fractional** **integral** operator given by (2.7) satisﬁes the inequality

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The necessary details of **fractional** **Hadamard** calculus are given in Kilbas [14] and Samko et al. [15]. Here we present some definitions of **Hadamard** derivative and **integral** as given in [2]. Definition 2.1. The **Hadamard** **fractional** **integral** of order α ∈ R + of function f (x), for all x > 1 is defined as,

From above definitions, we see the difference between **Hadamard** and Riemann-liouville **fractional** integrals as Kernel in the **Hadamard** **integral** has the from of ln( x t ) instead of the form of (x − t), which is involves in the Riemann- Liouville **integral**. The **Hadamard** derivative has the operator (x dx d ) n , while the Riemann-Liouville derivative has the operator ( d

Recently many authors have studied **integral** inequalities on **fractional** calculus using Riemann-Liouville, Caputo derivative, see [3, 5, 6, 7, 8, 9, 10]. The necessary background details are given in the book A.A. Kilbas [1], and in book of S.G. Samko et al. [4], here we present some definitions of **Hadamard** derivative and **integral** as given in [2, p.159-171].

**Fractional** calculus is a very helpful tool to perform diﬀerentiation and integration of real or complex number orders. This subject has earned much attention from researchers and mathematicians during the last few decades (see, e.g., [–]). Among a large number of the **fractional** **integral** operators developed, due to applications in many ﬁelds of sciences, the Riemann-Liouville **fractional** **integral** operator and **Hadamard** **fractional** **integral** op- erator have been extensively investigated.

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(b − t) 1−α dt. (4) The **fractional** **integral** in (3) coincides with the Riemann-Liouville **fractional** **integral** (1) when a = 0 and α = 1. It also coincides with the **Hadamard** **fractional** **integral** [9] once a = 0 and α → 0 with the Katugampola **fractional** **integral** [8], when a = 0. Similarly, Notice that, (Qf )(t) = f(a + b − t) then we have β a J α f (x) = Q( β J α b )f (x). Moreover (4)

Another kind of **fractional** derivative that appears in the literature is the **fractional** derivative due to **Hadamard**, introduced in [], which diﬀers from the Riemann- Liouville and Caputo derivatives in the sense that the kernel of the **integral** contains a log- arithmic function of an arbitrary exponent. Details and properties of **Hadamard** **fractional** derivative and **integral** can be found in [–]. Recently in the literature there appeared some results on **fractional** **integral** inequalities using the **Hadamard** **fractional** **integral**; see [–].

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**Fractional** calculus, as a very useful tool, shows its signiﬁcance to implement diﬀerenti- ation and integration of real or complex number orders. This topic has attracted much at- tention from researchers during the last few decades. Among a lot of the **fractional** **integral** operators that appeared, because of applications in many ﬁelds of sciences, the Riemann– Liouville **fractional** **integral** operator and **Hadamard** **fractional** **integral** operator have been extensively studied.

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In this paper, Hermite-**Hadamard** type inequalities involving **Hadamard** **fractional** integrals for the functions satisfying monotonicity, convexity and s-e-condition are studied. Three classes of left-type **Hadamard** **fractional** **integral** identities including the ﬁrst-order derivative are ﬁrstly established. Some interesting Hermite-**Hadamard** type **integral** inequalities involving **Hadamard** **fractional** integrals are also presented by using the established **integral** identities. Finally, some applications to special means of real numbers are given.

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Additionally, these hypotheses are valid in our results. Namely, if we get g(x) = x in (23), the inequality returns to (11). Similarly, getting g(x) = x in (24) gives inequality (10). Corollary 2 Hermite–**Hadamard** inequality for a B-convex function involving **Hadamard** **fractional** **integral** is obtained from inequalities (23) and (24).

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In this paper, we have obtained a new **fractional** **integral** identity, which played a key role in proving our main inequalities. Several Hermite–**Hadamard** type **fractional** **integral** in- equalities presented here, being very general, are pointed out to be specialized to yield some known results.

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Nowadays the **fractional** calculus has an important role in diverse scientiﬁc ﬁelds due to its several applications in dynamical problems including signals, hydrodynamics, dynamics, ﬂuid, viscoelastic theory, biology, control theory, image processing, computer network- ing, and many others [1–5]. A large number of scientists have worked on generalizations of existing results including theorems, deﬁnitions, models, and many more. A generaliza- tion of classical inequalities by means of **fractional**-order **integral** operators is considered as an interesting subject area. For instance, recently, Agarwal et al. [6] proved Hermite– **Hadamard**-type inequalities by using generalized k-**fractional**-integrals. Aldhaifallah et al. [7] used the (k, s)-**fractional** **integral** operator to generalize the inequalities for a fam- ily/class of n positive functions. Set et al. [8] studied Hermite–**Hadamard**-type inequalities for a generalized **fractional** **integral** operator for functions with convex absolute values of derivatives. Khan et al. [9] produced the Minkowski inequality by using the Hahn **integral** operator. On the other hand, noninteger-order calculus, usually referred to as **fractional** calculus, is used to generalize integrals and derivatives, in particular, integrals involving inequalities. Recently, Dumitru and Arran [10] introduced a new formula for **fractional** derivatives and integrals by using the Mittag-Leﬄer kernel. More theoretical concepts re- garding **fractional** operators with Mittag-Leﬄer kernels (Atangana–Baleanu operators) and the higher-order case have been discussed in [11, 12], whereas the generalization to the generalized Mittag-Leﬄer kernels to gain a semigroup property have been recently ini- tiated in [13, 14]. Khan [15] studied inequalities for a class of n functions by means of Saigo **fractional** calculus. Jarad et al. [16] presented a Gronwall-type inequality for the analysis of the **fractional**-order Atangana–Baleanu diﬀerential equation and in [17] for generalized **fractional** derivatives.

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**Fractional** analysis can be regarded as an expansion of classical analysis. **Fractional** anal- ysis has been studied by many scientists and they have expressed the **fractional** derivative and **integral** in diﬀerent ways with diﬀerent notations. Although the expressions between these diﬀerent deﬁnitions can be transformed into each other, but these diﬀerent deﬁni- tions and expressions have diﬀerent physical meanings. It is well known that the ﬁrst frac- tional **integral** operator is the Riemann–Liouville **fractional** **integral** operator. Recently, some new deﬁnitions of the **fractional** derivative were given by many mathematicians, which are the natural extensions of the classical derivative. These new deﬁnitions drew attention with their variability to classical derivative.

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Motivated by the work in [–], ﬁrstly, we will prove a generalization of the identity given by Zhu et al. using generalized **fractional** **integral** operators. Then we will give some new Hermite-**Hadamard** type inequalities, which are generalizations of the results in [] to the case λ = α, σ () = and w = . Our results can be viewed as a signiﬁcant extension and generalization of the previously known results.

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In this paper, we study the existence of solutions for a **fractional** boundary value problem involving **Hadamard**-type **fractional** diﬀerential inclusions and **integral** boundary conditions. Our results include the cases for convex as well as non-convex valued maps and are based on standard ﬁxed point theorems for multivalued maps. Some illustrative examples are also presented.

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This double inequality (1.1) is known in the literature as Hermite–**Hadamard** in- tegral inequality for convex functions. Both inequalities hold in the reversed di- rection if f is concave.The inequality (1.1) has been extended and generalized for various classes of convex functions via different approaches, see [4, 7, 10, 12]. For several recent result concerning the inequality (1.1) we refer the interested reader to [1 − 11, 13, 15 − 17, 19] and references cited therein.

The advantages of **fractional** calculus have been described and pointed out in the last few decades by many authors. **Fractional** calculus is based on derivatives and integrals of **fractional** order, **fractional** differential equations and methods of their solution. The most celebrated inequality has been studied extensively since it is established, is the Hermite-**Hadamard** inequality not only established for classical integrals but also for **fractional** integrals, see [18, 20, 27, 29].

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In this study some of the general versions of **Hadamard** inequality are analyzed in **fractional** calculus. A generalization of convex functions; namely (h − m)−convex function is used to establish these results. Some identities have been established which are further utilized in the formation of **Hadamard** type inequalities. Furthermore, **Hadamard** type inequalities for product of two (h − m)−convex functions have been studied and connection with already published results is investigated.

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In this work, we have established new conformable **fractional** **integral** inequalities of Hermite–**Hadamard** type for convex functions using the. As a special case, if we substitute α = 1 into the general definition of conformable **fractional** integrals (Definition 2), we obtain the classical integrals. In view of this, we obtained some new inequalities of Hermite–**Hadamard** type for convex functions involving classical integrals.

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