ON FENG QI-TYPE INTEGRAL INEQUALITIES FOR CONFORMABLE FRACTIONAL INTEGRALS

In this paper, we establish the generalized Qi-type inequality involving conformable fractional integrals. The results presented here would provide extensions of those given in earlier works. 1. Introduction In the last few decades, much signicant development of integral inequalities had been established. Integral inequalities have been frequently employed in the theory of applied sciences, di¤erential equations, and functional analysis. In the last two decades, they have been the focus of attention in [4]-[7]. Recently, especially Qi inequality, one of the integral inequalities, has been studied by many authors. Recall the famous integral inequality of Feng Qi type: Theorem 1. (Proposition 1.1, [7]) Let f(x) be di¤erentiable on (a; b) and f(a) = 0. If f 0 (x) 1 for x 2 (a; b), then (1.1) 0@ b Z a [f (t)] 3 dt 1A 0@ b Z a f (t) dt 1A2 : If 0 f 0 (x) 1, then the inequality (1.1) reverses. Theorem 2. (Proposition 1.3, [7]) Let n be a positive integer. Suppose f(x) has continuous derivative of the n th order on the interval [a; b] such that f (i) (a) 0, where 0 i n 1, f (n) (x) n!; then (1.2) 0@ b Z a [f (t)] n+2 dt 1A 0@ b Z a f (t) dt 1An+1 : In [6], Ngô et al. gave the following inequality which is one of the open problems solution. Theorem 3. If f is a nonnegative, continuous function on [0; 1] satisfying, (1.3) 1 Z x f (t) dt 1 x 2 2 ; Key words and phrases. Integral Inequalities, Special Functions, Fractional Calculus, Conformable Fractional Integral. 2010 Mathematics Subject Classication. 26A33, 26D10, 26D15, 41A55. 1 Department of Mathematics, Faculty of Science and Arts, University of KahramanmaraŞ Sütçü · Imam, 46100, KahramanmaraŞ, Turkey E-mail address : abdullahmat@gmail.com Department of Mathematics, Faculty of Science, University of Cumhuriyet, 58140, Sivas, Turkey E-mail address : mesra@cumhuriyet.edu.tr Department of Mathematics, Faculty of Science and Arts, University of KahramanmaraŞ Sütçü · Imam, 46100, KahramanmaraŞ, Turkey E-mail address : hyildir@ksu.edu.tr Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 27 September 2016 doi:10.20944/preprints201609.0105.v1


Introduction
In the last few decades, much signi…cant development of integral inequalities had been established. Integral inequalities have been frequently employed in the theory of applied sciences, di¤erential equations, and functional analysis. In the last two decades, they have been the focus of attention in [4]- [7]. Recently, especially Qi inequality, one of the integral inequalities, has been studied by many authors. Recall the famous integral inequality of Feng Qi type: In  (1.4) holds.
Lemma 1. (General Cauchy inequality). Let and be positive real numbers satisfying + = 1. Then for all positive real numbers x and y, we have x + y x y :

Definitions and properties of conformable fractional derivative and integral
The following de…nitions and theorems with respect to conformable fractional derivative and integral were referred in [1], [9]- [13].
for all t > 0; 2 (0; 1) : If f is di¤ erentiable in some (0; a) ; > 0; lim We can write f ( ) (t) for D (f ) (t) to denote the conformable fractional derivatives of f of order . In addition, if the conformable fractional derivative of f of order exists, then we simply say f is di¤ erentiable.
The idea of derivative of non-integer order was motivated by the question, "What does it mean by d n f dx n , if n = 1 2 ?", asked by L'Hospital in 1695 in his letters to Leibniz ([15]- [17]). Afterwards, many mathematicians tried to answer this question for centuries in several points of view. Various types of fractional derivatives were introduced by many authors, most of them are de…ned via fractional integrals, but many of those fractional derivatives have some non-local behaviors. Among the inconsistencies of the existing fractional derivatives are:

3
(2) All fractional derivatives do not obey the familiar Product Rule and Quotient Rule for two functions.
(3) All fractional derivatives do not obey the Chain Rule.
(4) Fractional derivatives do not have a corresponding Rolle's Theorem, Mean Value Theorem.
(5) All fractional derivatives do not obey: D a D a f = D a + f , in general. To solve some of these and other di¢ culties, Khalil et al. [10], introduced the following. iii De…nition 2 (Conformable fractional integral). Let 2 (0; 1] and 0 a < b: exists and is …nite.
where the integral is the usual Riemann improper integral, and 2 (0; 1]. Theorem 8. Let f : (a; b) ! R be di¤ erentiable and 0 < 1. Then, for all t > a we have Theorem 9. (Integration by parts) Let f; g : [a; b] ! R be two functions such that f g is di¤ erentiable. Then The simple nature of this de…nition allows for many extensions of some classical theorems in calculus for which the applications are indispensable in the fractional di¤erential models that the existing de…nitions do not permit.

Main Results
We start the following important inequality for conformable fractional integrals: Simple computation yields Since f 0 (t) 0 and f (a) = 0, thus f (t) is increasing and f (t) 0.
By using our assumption we have On the other hand, integrating by parts, we also get The proof is completed.
Integrating by parts, we have On the other hand, by using our assumption we have The proof is completed. Proof. Integrating by parts, we have On the other hand, by using our assumption we have Thus, The proof is completed.
holds for every positive real number n > 0; m > 0 and > 0.
Proof. By using the Cauchy inequality, we obtain m n + m + 1 x n+ 1 f m (x) d x: which completes this proof.

Conclusion 1.
In the present paper, we establish the generalized Qi-type inequalities involving conformable fractional integrals. Some special cases are also discussed.