ON CAPUTO FRACTIONAL DERIVATIVES VIA CONVEXITY

. In this paper some estimations of Caputo fractional derivatives via convexity have been presented. By using convexity of any positive integer order diﬀerentiable function some novel results are given.

If α = n ∈ {1, 2, 3, . . . } and usual derivative f (n) (x) of order n exists, then Caputo fractional derivative C D n a+ f (x) coincides with f (n) (x) whereas C D n b− f (x) coincides with f (n) (x) with exactness to a constant multiplier (−1) n . In particular we have , where n = 1 and α = 0.
Since the inequalities always have been proved worthy in establishing the mathematical models and their solutions in almost all branches of applied sciences (see [2,3]). Especially the convexity takes very important role in the optimization theory. The aim of this paper is to introduce some fractional inequalities for the Caputo fractional derivatives via the convexity property of the functions which have derivatives of any integer order.

Main Results
First we give the following estimate of the sum of left and right handed Caputo fractional derivatives.
Proof. Let us consider the function f on the interval [a, x], x ∈ [a, b]. For t ∈ [a, x], the following inequality holds Since f (n) is convex therefore for t ∈ [a, x] we have

Multiplying inequalities (2.3) and (2.2), then integrating with respect to t over [a, x]
we have Multiplying inequalities (2.5) and (2.6), then integrating with respect to t over [ Adding (2.4) and (2.7) we get the required inequality in (2.1).
It is nice to see that the following implication holds.
Corollary 2.1. By setting α = β in (2.1) we get the following fractional integral inequality Now we give the next result stated in the following theorem.
We consider the second inequality of inequality (2.9) The product of last two inequalities give Integrating with respect to t over [a, x] we have . Therefore, (2.11) takes the form If one consider from (2.9) the first inequality and proceed as we did for the second inequality, then following inequality can be obtained From (2.12) and (2.13) we get On the other hand for t ∈ [x, b] using convexity of |f (n+1) | we have Also for t ∈ [x, b] and β > 0 we have By adopting the same treatment as we have done for (2.9) and (2.10) one can obtain from (2.15) and (2.16) the following inequality By combining the inequalities (2.14) and (2.17) via triangular inequality we get the required inequality.
It is interesting to see the following inequalities as a special case.

Corollary 2.2.
By setting α = β in (2.8) we get the following fractional integral inequality Before going to the next theorem we observe the following result.
Proof. We have Since f is convex, therefore we have − x)) .
Also f is symmetric about a+b 2 , therefore, we have f (a + b − x) = f (x) and inequality in (2.18) holds.
Multiplying with (x − a) n−β on both sides and then integrating over [a, b] we have By definition of Caputo fractional derivatives one can has Multiplying (2.25) with (b − x) n−α , then integrating over [a, b] one can get Adding (