Hermite–Hadamard type inequalities for fractional integrals via Green’s function

In the article, we establish the left Riemann–Liouville fractional Hermite–Hadamard type inequalities and the generalized Hermite–Hadamard type inequalities by using Green’s function and Jensen’s inequality, and present several new Hermite–Hadamard type inequalities for a class of convex as well as monotone functions.


Introduction
Convexity plays an important role in all the fields of pure and applied mathematics [1][2][3][4][5][6][7][8][9][10][11][12]. Many remarkable inequalities have been obtained in the literature by using convexity [13][14][15][16][17][18][19][20][21][22]. Among the inequalities, the most extensively and intensively attractive inequality in the last decades is the well-known Hermite-Hadamard inequality. This interesting result was obtained by Hermite and Hadamard independently, and it provides an equivalence with the convexity property. This inequality reads as follows: if the function ψ : [α 1 , α 2 ] → R is convex on [α 1 , α 2 ], then If ψ is a concave function, then the inequalities in (1.1) will hold in reverse directions. The Hermite-Hadamard inequality gives an upper as well as lower estimations for the integral mean of any convex function defined on a closed and bounded interval which involves the endpoints and midpoint of the domain of the function. Also (1.1) provides the necessary and sufficient condition for the function to be convex. There are several applications of this inequality in the geometry of Banach spaces and nonlinear analysis [23,24]. Some peculiar convex functions can be used in (1.1) to obtain classical inequalities for means. For some comprehensive surveys on various generalizations and developments of (1.1), we recommend [25]. Due to the great importance of this inequality, in the recent years many remarkable varieties of generalizations, refinements, extensions and different versions of Hermite-Hadamard inequality for different classes of convexity, such as preinvex, s-convex, harmonic convex, α(x)-convex, superquadratic, and co-ordinate convex functions, have been studied in the literature. Also there have been a large number of research papers published on this subject, for interested readers we recommend to read the papers [26][27][28][29][30][31][32][33][34][35][36][37] and some of the references therein. The following definitions for the left and right side Riemann-Liouville fractional integrals are well known in the literature.
Then the left and right Riemann-Liouville fractional integrals J α b 1 + ψ and J α b 2 -ψ of order α > 0 are defined by respectively, where (α) is the gamma function defined by (α) = ∞ 0 e -t t α-1 dt. In [38], Sarikaya et al. established the Hermite-Hadamard type inequality for fractional integral as follows.
Remark 1.2 In Theorem 1.1, it is not necessary to suppose that ψ is a positive function and b 1 , b 2 are positive real numbers. From the definition of left and right Riemann-Liouville fractional integrals, we clearly see that b 1 and b 2 can be any real numbers such that b 1 < b 2 .
The main purpose of this paper is to give a new method to derive the left Riemann-Liouville fractional Hermite-Hadamard type inequalities as given in [39]. In this method we use Green's function and obtain identities for the difference of the left Riemann-Liouville fractional Hermite-Hadamard inequality, and then we prove that these identities are non-negative. As a consequence, these inequalities provide the generalized Hermite-Hadamard inequality. Also, by using these identities for the class of convex, concave, and monotone functions, we obtain new Hermite-Hadamard type inequalities.

Main results
Let b 1 < b 2 . Then the following four new Green's functions 3,4) are defined by Mehmood et al. in [40]: In [40], the authors established the following Lemma 2.1, which will be used to establish our main results. .
) be a convex function. Then the double inequality holds for any α > 0.
Next, we present new Hermite-Hadamard type inequalities for the class of monotone and convex functions.
) and α > 0. Then the following statements are true: (i) If |ψ | is an increasing function, then (iii) If |ψ | is a convex function, then Proof (i) It follows from (2.15) that α+1 ≥ 0 and |ψ | is an increasing function, therefore we have which is inequality (2.17). Part (ii) can be proved in a similar way, we omit the details.
For part (iii), making use of (2.18) and the fact that every convex function ψ defined on the interval [b 1 , b 2 ] is bounded above by max{ψ(b 1 ), ψ(b 2 )}, we get Remark 2.4 Let α = 1. Then Theorem 2.3 leads to ) and α > 0. Then the following statements are true: (i) If |ψ | is an increasing function, then (ii) If |ψ | is a decreasing function, then (iii) If |ψ | is a convex function, then Proof (i) It follows from (2.9) that Taking the absolute function and using the triangular inequality, we get Part (ii) can be proved by using the same procedure.
Next, we prove part (iii). We clearly see that Since every convex function ψ defined on an interval [b 1 , b 2 ] is bounded above by max{ψ(b 1 ), ψ(b 2 )}. Therefore, we have which is our required inequality.
Remark 2.8 In Theorem 2.7, if we take α = 1, then we obtain
Proof From (2.15), we have Let t ∈ [0, 1] and μ = tb 1 + (1t)b 2 . Then we obtain Taking absolute on both sides and using the convexity of |ψ |, we get which is our required inequality.
Remark 2.12 In Theorem 2.11, if we take α = 1, then we obtain