On Some Inequalities Involving Liouville–Caputo Fractional Derivatives and Applications to Special Means of Real Numbers

We are concerned with the class of functions f ∈ C1([a, b];R), a, b ∈ R, a < b, such that |cDα a f | is convex or ∣∣cDα b f ∣∣ is convex, where 0 < α < 1, cDα a f is the left-side Liouville–Caputo fractional derivative of order α of f and cDα b f is the right-side Liouville–Caputo fractional derivative of order α of f . Some extensions of Dragomir–Agarwal inequality to this class of functions are obtained. A parallel development is made for the class of functions f ∈ C1([a, b];R) such that |cDα a f | is concave or ∣∣cDα b f ∣∣ is concave. Next, an application to special means of real numbers is provided.


Introduction
Let 0 < α < 1 and a, b ∈ R be such that a < b. We consider the classes of functions where c D α a f is the left-side Liouville-Caputo fractional derivative of order α of f and c D α b f is the right-side Liouville-Caputo fractional derivative of order α of f . In this paper, we extend Dragomir-Agarwal inequality to the above classes of functions. Next, we provide an application to the special means of real numbers.
Let us mention some motivations for studying the proposed problems. Let f : I → R be a given function, where I is a certain interval of R, and let a, b ∈ I be such that a < b. If f is convex in I, then Inequality (5) is known in the literature as Hermite-Hadamard's inequality (see [1,2]). Several improvements and extensions of inequality (5) to different types of convexity were established by many authors. In this direction, we refer the reader to [2][3][4][5][6][7][8] and the references therein. In [9], Dragomir and Agarwal established the following interesting result, which provides an estimate between the difference between the middle and right terms in inequality (5).
Theorem 1 (Dragomir-Agarwal inequality). Let f : I → R be a given function, where I is a certain interval of R, and let a, b ∈ I be such that The main idea for proving Theorem 1 is based on the following lemma [9].
Lemma 1. Let f : I → R be a given function, where I is a certain interval of R, and let a, b ∈ I be such that In [10], Pearce and Pečarić extended Theorem 1 to the case when | f | is concave. Using Lemma 1 and Jensen integral inequality, they obtained the following interesting result.
Theorem 2. Let f : I → R be a given function, where I is a certain interval of R, and let a, b ∈ I be such that Motivated by the above cited works, our aim in this paper is to extend Theorems 1 and 2 to the classes of functions given by (1)- (4).
The rest of the paper is organized as follows. In Section 2, we recall some basic concepts on fractional calculus. In Section 3, we state and prove our main results. In Section 4, an application to special means of real numbers is provided.

Preliminaries
In this section, we recall some basic notions on fractional calculus. For more details, we refer the reader to [11,12].
First, let us fix a, b ∈ R with a < b.
where Γ denotes the Gamma function.
Lemma 2. Let σ > 0 and f ∈ C([a, b]; R). Then, The following result is an immediate consequence of Lemma 2.
The following result is an immediate consequence of Lemma 3.
The following result provides sufficient conditions for the convexity and concavity of c D α a f (see [13]).
Using Lemma 7, we deduce the following criteria for the convexity and concavity of | c D α a f |.
Proof. We have just to observe that, by (7), we have Next, using Lemma 7, the desired results follow.

Results and Discussion
In this section, we state and prove our main results. Just before, let us fix 0 < α < 1 and a, b ∈ R with a < b.
First, we shall establish the following fractional version of Lemma 1.
Proof. By the definition of the left-side Liouville-Caputo fractional derivative of order α, we have

Using the integration by parts rule given by Lemma 4, we obtain
Next, the standard integration by parts rule yields b a x − a+b On the other hand, by the definition of the right-side Riemann-Liouville fractional integral of order 1 − α, we have Using the identity sΓ(s) = Γ(s + 1), s > 0, we obtain and Using (11), we obtain b a f (x) d dx Next, combining (9), (10) and (12), we obtain Finally, the change of variable Remark 1. Passing to the limit as α → 1 − in (8) and using Lemmas 3 and 6, we obtain (6).
Using a similar argument as in the proof of Lemma 9, we obtain the following fractional version of Lemma 1.
Our first main result is the following fractional version of Theorem 1.
is the class of functions given by (1). Then, Proof. Let f ∈ Conv α a ([a, b]; R). Using Lemma 9, we obtain On the other hand, using the convexity of | c D α a f |, we obtain Using the fact that ( c D α a f )(a) = 0 and Finally, combining (15) and (16), we obtain (14).
Next, we discuss the case when is the class of functions given by (2).
Proof. Using Lemma 10, the convexity of | c D α b f | and a similar argument as in the proof of Theorem 3, we obtain (17).
Furthermore, we consider the case when f ∈ Conc α a ([a, b]; R), where Conc α a ([a, b]; R) is the class of functions given by (3). We obtain the following fractional version of Theorem 2.
Proof. Using the concavity of | c D α a f | and Jensen integral inequality, we obtain Using Lemma 9 and the above inequality, (18) follows.
Using a similar argument as in the proof of Theorem 5, we obtain the following result concerning the case when is the class of functions given by (4).
To show this, let us consider as example the function where α + 1 ≤ β < 2.
We have f ∈ C 1 ([a, b]; R). Moreover, It can be easily seen that | f | is concave (so, nonconvex). On the other hand, we have which is a convex function in [a, b]. Therefore, f ∈ Conv α a ([a, b]; R) but f ∈ Conv 1 ([a, b]; R). Hence, Theorem 3 is a real extension of Theorem 1.

Applications to Special Means of Real Numbers
In this section, we provide some applications to special means of real numbers. Let The quantity A(u, v) is known in the literature as the arithmetic mean of u and v. Let The quantity L σ (u, v) can be considered as a fractional generalized ln-mean of u and v. We have the following estimate. Corollary 1. Let 1 ≤ β ≤ 2 and 0 < a < b. Then,