Generalized inequalities involving fractional operators of the Riemann-Liouville type

1 Facultad de Ingenierı́a, Universidad del Desarrollo, Ave. La Plaza 680, San Carlos de Apoquindo, Las Condes, Santiago 7550000, Chile 2 Centro Acapulco, Facultad de Matemática, Universidad Autónoma de Guerrero, Acapulco de Juárez, Guerrero 39610, Mexico 3 Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, Leganés, Madrid 28911, Espaa 4 Centro Acapulco, Facultad de Matemática, Universidad Autónoma de Guerrero, Acapulco de Juárez, Guerrero 39610, Mexico


Introduction
The idea of fractional calculus is as old as traditional calculus (see [1]). Until recently, research on fractional calculus was confined to the field of pure mathematics but, in the last two decades, many applications of fractional calculus appeared in several fields of engineering, applied sciences, physics, economy, etc.
For a complementary study on the recent developments in the field of fractional calculus as well as its applications see [2][3][4][5][6][7].
It is important to note that the global fractional derivatives (e.g., Caputo and Riemann-Liouville) are not collecting mere local information. By contrast, fractional operators keep track of the history of the process being studied; this feature allows modeling the non-local and distributed responses that commonly appear in natural and physical phenomena. On the other side, one has to recognize that these fractional derivatives D α show some drawbacks.
This paper relies on the introduction and use of new differential operators, depending on a general kernel function, which include at once several fractional derivatives earlier introduced and studied in many different sources.
As we know, by manipulating simple algebraic identities, we can follow the idea of fractional differential operators of Riemann-Liouville or Caputo type [8][9][10]. In this paper we will use a general kernel T in order to define general integral and differential operators of Riemann-Liouville type: We state the main properties of these integral operators. Furthermore, we study Ostrowski, Székely-Clark-Entringer and Hermite-Hadamard-Fejér inequalities involving these general fractional operators.

Preliminaries
One of the first operators that can be called fractional is the Riemann-Liouville fractional derivative of order α ∈ C, with Re(α) > 0, defined as follows (see [11]). Definition 1. Let a < b and f ∈ L 1 ((a, b); R). The right and left side Riemann-Liouville fractional integrals of order α, with Re(α) > 0, are defined, respectively, by with t ∈ (a, b).
When α ∈ (0, 1), their corresponding Riemann-Liouville fractional derivatives are given by Other definitions of fractional operators are the following ones.
Definition 2. Let a < b and f ∈ L 1 ((a, b); R). The right and left side Hadamard fractional integrals of order α, with Re(α) > 0, are defined, respectively, by
In [12], the author introduced new fractional integral operators, called the Katugampola fractional integrals, in the following way.
→ R an integrable function, and α ∈ (0, 1), ρ > 0 two fixed real numbers. The right and left side Katugampola fractional integrals of order α are defined, respectively, by Some generalizations of the Riemann-Liouville and Hadamard fractional derivatives appeared in [13]. These generalizations, called Katugampola fractional derivatives, are defined as The relations between these two fractional operators are the following: respectively (see [14]), by

7)
and There are other definitions of integral operators in the global case, but they are slight modifications of the previous ones, some include non-singular kernel and others incorporate different terms.

General fractional integral of Riemann-Liouville type
Now, we give the definition of a general fractional integral.
The right and left integral operators, denoted respectively by J α T,a + and J α . Note that these operators generalize the integral operators in Definitions 1-4: then J α T,a + and J α T,b − are the right and left Riemann-Liouville fractional integrals RL J α a + and RL J α b − in (2.1) and (2.2), respectively. Its corresponding right and left Riemann-Liouville fractional derivatives are (B) If we choose (C) If we choose then J α T,a + and J α T,b − are the right and left Katugampola fractional integrals K α,ρ a + and K α,ρ b − in (2.5) and (2.6), respectively. Its corresponding right and left Katugampola fractional derivatives are (D) If we choose a function g with the properties in Definition 5 and then J α T,a + and J α T,b − are the right and left Kilbas-Marichev-Samko fractional integrals I α g,a + and I α g,b − in (2.7) and (2.8), respectively.
Definition 6. Let a < b and α ∈ R + . Let g : [a, b] → R be a positive function on (a, b] with continuous positive derivative on (a, b), and G : , its right and left generalized derivative of order α are defined, respectively, by for each t ∈ (a, b).
Note that if we choose Also, we can obtain Hadamard and Katugampola fractional derivatives as particular cases of this generalized derivative.

Properties of the integral operators
The following result collects some elementary properties of J α T,a + and J α T,b − .
(2) For every functions f 1 , (3) For every function f ∈ L 1 T [a, b] and t ∈ [a, b], we have ds. Proof. By using Hölder inequality, since T (t, s, α) ≥ 0, we have for each t ∈ [a, b] In a similar way, since  Proof. Denote by I B the characteristic function of the set B (i.e., the function such that I B (t) = 1 if t ∈ B and I B (t) = 0 if t B). Then Therefore, J α T,a + is a Hilbert-Schmidt integral operator on L 2 [a, b], thus, it is a linear compact operator. This finishes the proof of the first item. The proof of the second one is similar.

On the Ostrowski inequality in the generalized framework
The utility of inequalities, particularly integral inequalities involving convex functions, is widely recognized as one of the main elements supporting the development of several modern branches of mathematics, and so, it has received considerable attention in recent years.
Ostrowski proved in [15] the following interesting inequality: Since then, there are a lot of generalizations and applications of this inequality (see, e.g., [16]). In particular, Dragomir and Wang generalized this inequality to L p [a, b] (p > 1) in [17] as follows: Theorem 12. Let f : [a, b] → R be a differentiable function. If p > 1, 1/p + 1/q = 1 and f ∈ L p [a, b], then In this paper we prove a version of this inequality involving our kernel 1/T . The main improvement is to consider this general weight, but also, we prove the inequality for a larger class of functions, and we include the case p = 1. (1) If 1 < p ≤ ∞ and 1/p + 1/q = 1, then (2) If p = 1, then Proof. First of all, let us check that Assume first 1 < p < ∞. Hölder inequality gives The desired inequality holds since If p = 1 or p = ∞, then a similar and simpler argument gives the inequalities.

On the Székely-Clark-Entringer inequality
The following Székely-Clark-Entringer inequality appears in [18]. In this section we are going to prove a Székely-Clark-Entringer-type inequality for generalized integrals.
Theorem 15. Consider real numbers a < b, α > 0, 0 < r ≤ p and f : [a, b] → R a measurable function. Then the following inequality for fractional integrals holds: The argument in the proof of Theorem 15 allows to obtain a strongly improvement of Proposition 14 (it appears in [19], with a more complicated proof). for any measure µ and 0 < r ≤ p. In particular, since (x 1 , . . . , x n ) ∞ ≤ ∆, we have n j=1 x j ≤ ∆ 1−r n j=1 x r j .
Thus, the following known inequality

On the Hermite-Hadamard inequality
The following double inequality holds for any convex function f on [a, b].
This inequality was published by Hermite in 1883 and, independently, by Hadamard in 1893. It gives an estimation of the mean value of a convex function and note that it also provides a refinement of Jensen inequality. Probably the most important extension of this inequality is the so called Hermite-Hadamard-Fejér inequality for any convex function f on [a, b] and any non-negative integrable function g which is symmetric with respect to (a + b)/2. The motivated reader is referred to [20] and references therein for more information and other extensions of Hermite-Hadamard inequality.
In [21] the authors proved the following variant of Hermite-Hadamard inequality for Riemann-Liouville fractional integrals: Since µ is a symmetric measure with respect to (a + b)/2, we have