SOME GENERAL INTEGRAL INEQUALITIES FOR LIPSCHITZIAN FUNCTIONS VIA CONFORMABLE FRACTIONAL INTEGRAL

In this paper, the author establishes some Hadamard-type and Bullen-type inequalities for Lipschitzian functions via Riemann Liouville fractional integral.


Introduction
Hermite-Hadamard Inequality. Let f : I R ! R be a convex function de…ned on the interval I of real numbers and a; b 2 I with a < b. The following inequality holds. This double inequality is known in the literature as Hermite-Hadamard integral inequality for convex functions (see [7]). Note that some of the classical inequalities for means can be derived from (1) for appropriate particular selections of the mapping f . Ostrowski's Inequality. Let f : I R ! R be a mapping di¤erentiable in I ; the interior of I, and let a; b 2 I with a < b: If jf 0 (x)j M; x 2 [a; b] ; then we the following inequality holds for all x 2 [a; b] (see [1]).
Bullen's inequality. Suppose that f : [a; b] ! R is a convex function on [a; b]. Then we have the inequalities: (see [5] and [16]). In what follows we recall the following de…nition.
for all x; y 2 I: For some recent results are connected with Hermite-Hadamard type integral inequalities for Lipschitzian functions, see [4,8,9,17,18]. In [17], Tseng et al. established some Hadamard-type and Bullen-type inequalities for Lipschitzian functions as follows: Theorem 2. Let I be an interval in R, a A B b in I, V = (1 )a + b, 2 [0; 1] and let f : I! R be an L -Lipschitzian function with L 0: Then we have the inequality :   954   ·  IM DAT ·  IŞCAN, SERCAN TURHAN, AND SELIM NUM AN   Theorem 3. Let I be an interval in R, a A B C b in I, V 1 = (1 )a+ b, V 2 = a + ( + ) b, ; ; 2 [0; 1], + + = 1, and let f : I ! R be an L-Lipschitzian function with L 0: Then we have the inequality where V ; ; is de…ned as in [17,Section 3].
We give some necessary de…nitions and mathematical preliminaries of fractional calculus theory which are used throughout this paper. [13]). In the case of = 1, the fractional integral reduces to the classical integral. For some recent results connected with fractional integral inequalities, see [2,10,14,15,19]. In [15] with > 0: De…nition 6. Let 2 (n; n + 1] ; n = 0; 1; 2; ::: and set = n. Then the left conformable factional integral of any order > 0 is de…ned by and analogously, the right conformable fractional integral of any order > 0 is de…ned by Notice that, if = n + 1 then = n = 1 and hence (I a f ) (x) = J n+1 a+ f (x) : Also, if n = 0 and = 1 then = 1 and hence The Beta function de…ned as follows: The Incomplete Beta function is de…ned by for x = 1, the incomplete beta function coincides with the complete beta function.
In [12], Set et. al. represented Hermite-Hadamard's inequalities for conformable fractional integrals as follows: ]. If f is a convex function on [a; b], then the following inequalities for conformable fractional integrals hold: The aim of this paper is to indicate generalizations of some integral inequalities for Lipschitzian functions via conformable fractional integral. The results are obtained in this study is a generalization of the results which are obtained in Theorem 2 and Theorem 3 by using conformable fractional integrals.

A generalization of Hadamard and Ostrowski type inequalities for Lipschitzian functions via fractional integrals
Throughout this section, let I be an interval in R, a x y b in I and let f : I ! R be an M -Lipschitzian function. In the next theorem, let 2 [0; 1], A = (1 )a + b, and A ; ;n , > 0, n = 0; 1; 2, = n; as follows: K ; ;n (x; y; a) = 0; Theorem 8. Let x; y; ; ; A; A ; ;n and the function f be de…ned as above. Then we have the inequality for fractional integrals Proof. Using the hypothesis of f , we have the following inequality Now using simple calculations, we obtain the following identities  Using the inequality (7) and the above identities R A a jx tj (A t) n (t a) 1 dt and R b A jy tj (t A) n (b t) 1 dt, we derive the inequality ( 6). This completes the proof.
Under the assumptions of Theorem 8, we have the following corollaries and remarks as follows: Remark 9. In Theorem 8, if we take = = 1 and n = 0, then the inequality (6) reduces the inequality (2) in Theorem 2 under the appropriate symbols.

A generalization of Bullen and Simpson type inequalities for Lipschitzian functions via fractional integrals
Throughout this section, let I be an interval in R, a x y z b in I and f : I ! R be an M -lipschitzian function. In the next theorem, let + + = 1, ; ; 2 [0; 1], A = (1 )a + b, C = a + ( + ) b, and de…ne I ; ; ; , > 0, as follows: (1) If A C x y z or A x C y z, then Proof. Using the hypothesis of f , we have the inequality Now, using simple calculations, we obtain the following identities (1) If A C x y z or A x C y z, then we have ; (x; y; z): (2) If A x y C z, then we have Using the inequality (11) and the above identities R A a jx tj (A t) n (t a) 1 dt, R C A jy tj (t A) n (C t) 1 dt and R b C jz tj (t C) n (b t) 1 dt, we derive the inequality (10). This completes the proof.
Under the assumptions of Theorem 14, we have the following corollaries and remarks as follows: Remark 15. In Theorem 14, if we take = = 1 and n = 0, then then the inequality (10) reduces the inequality (3)