Monotonicity, Concavity, and Convexity of Fractional Derivative of Functions

The monotonicity of the solutions of a class of nonlinear fractional differential equations is studied first, and the existing results were extended. Then we discuss monotonicity, concavity, and convexity of fractional derivative of some functions and derive corresponding criteria. Several examples are provided to illustrate the applications of our results.

It is well known that the monotonicity, the concavity, and the convexity of a function play an important role in studying the sensitivity analysis for variational inequalities, variational inclusions, and complementarity. Since fractional derivative of a function is usually not an elementary function, its properties are more complicated than those of integer order derivative of the function. The focal point of this paper is to investigate the monotonicity, the concavity, and the convexity of fractional derivative of some functions. Now we recall some definitions and lemmas which will be used later. For more detail, see [1][2][3][4].

Definition 1. Given an interval [ , ]
of R, the fractional order integral of a function ∈ 1 [ , ] of order ∈ R + is defined by where Γ is the Gamma function.

Definition 2.
Riemann-Liouville's derivative of order with the lower limit for a function ∈ 1 [ , ] can be written as Definition 3. Suppose that a function is defined on the interval [ , ] and ( ) ( ) ∈ 1 [ , ]. The Caputo's fractional derivative of order with lower limit for is defined as where 0 < − 1 < ≤ .

Lemma 4. There exists a link between Riemann-Liouville and
Caputo's fractional derivative of order . Namely, where Re( ) denotes the real parts of . Particularly, for 0 < < 1, it holds that holds.
The rest of this paper is organized as follows. Section 2 is devoted to monotonicity of solutions of fractional differential equations. In Section 3, we present the monotonicity, the concavity, and the convexity of functions RL 0 ( ) and 0 ( ). Summarizing this paper forms the content of Section 4.

Monotonicity of Solutions of Nonlinear Fractional Differential Equations
In this section, we mainly investigate the monotonicity of the solution of nonlinear fractional differential equation with Caputo's derivative which was discussed in [6,21], where 0 < < 1. The paper [21]  In [21], the authors gave an improvement of Lemma 1.7.3, which is as follows.
Proof. The conclusion of (1) is obvious. In fact, (8) is equivalent to Now we prove the validity of (2) and (3). First, by the definition of the Caputo's derivative, it holds from (8) that Then it follows that That is, Then we can get thaṫ Similar to the proof of (2), we can prove that (3) holds. This completes the proof. [21] is a particular case of Theorem 6 of this paper. In fact in Lemma 2.4 in [21], if ( , ) = , then (8) is

Remark 7. Lemma 2.4 in
The Scientific World Journal which satisfies the conditions of (2) in Theorem 6.
The proof of (4) is similar to that of (3). This completes the proof. Now we are to investigate the monotonicity of the function 0 ( ). Theorem 13. Assume that 0 < < 1. If there exists an interval Proof. Set( ) = ( ). Note that The proof is completed.
The following examples illustrate applications of Theorems 12 and 13.  Next we are to investigate the concavity and the convexity of RL 0 ( ) and 0 ( ). By formula (23), we have The Scientific World Journal Thus we can obtain the following theorem.
Theorem 17. Assume that 0 < < 1. If there exists an interval The next theorem is on the convexity and the concavity of 0 ( ). Theorem 18. Assume that 0 < < 1. If there exists an interval If ( ) ≥ 0 on [ 0 , 1 ],( 0 ) ≥ 0, and( 0 ) ≤ 0, then Example 19. Assume that 0 < < 1. Consider the fractional differential equation . Now we employ the method which is used in the proof of Theorem 17 to investigate it. By formula (29), we have The three terms in the right side of (31) can be reduced to It is not difficult to get for ∈ [ 0 + ( + (4 − 3 2 ) 0.5 )/2, +∞) and

Conclusions
In this paper, we first investigate the monotonicity of solutions of nonlinear fractional differential equations with the Caputo's derivative. The results we derive are an improvement of the existing results. Meanwhile, several examples are provided to illustrate the applicability of our results.
The main part of this paper is to study the monotonicity, the concavity, and the convexity of the functions RL 0 ( ) and 0 ( ). Based on the relation between the Riemann-Liouville fractional derivative and the Caputo's derivative, we obtain the criteria on the monotonicity, the concavity, and the convexity of the functions RL 0 ( ) and 0 ( ). In the meantime, five examples are given to illustrate the applications of our criteria.