ON THE HERMITE-HADAMARD-MERCER TYPE INEQUALITIES FOR GENERALIZED PROPORTIONAL FRACTIONAL INTEGRALS

Our aim in this paper is to establish some new Hermite-HadamardMercer type integral inequalities by utilizing the fractional proportional-integral operators.For this purpose, Hermite-Hadamard-Mercer inequalities for di¤erantiable mappings whose derivatives in absolute value are convex. 1. INTRODUCTION AND PRELIMINARES Over the past few years, various researchers studied the so-called conformable integrals and derivatives. According to this idea, some authors used modied proportional derivatives to create nonlocal fractional integrals and derivatives, called fractional proportional integrals and derivatives, including see exponential functions in their kernels ([5] ; [7] ; [31] ; [32] ; [33]). Our purpose here new Hermite-HadamardMercer integral inequalities in the article some convex functions using fractional proportional integral operators. Let 0 < x1 x2 xn , = ( 1; 2; ; n) nonnegative and n P j=1 j = 1: The Jensen inequality [13] in literature states that if f is a convex function then,


INTRODUCTION AND PRELIMINARES
Over the past few years, various researchers studied the so-called conformable integrals and derivatives. According to this idea, some authors used modi…ed proportional derivatives to create nonlocal fractional integrals and derivatives, called fractional proportional integrals and derivatives, including see exponential functions in their kernels ( [5] ; [7] ; [31] ; [32] ; [33]). Our purpose here new Hermite-Hadamard-Mercer integral inequalities in the article some convex functions using fractional proportional integral operators.

Main Results
Using the Jensen-Mercer inequality, Hermite Hadamard inequalities can be represented in generalized proportional fractional integral forms as follows.
Theorem 2. Let f be a positive continuous, decreasing and convex function on the interval [ ; ], then the following inequality for, generalized proportional fractional integrals holds; (2.1) Then where 2 (0; 1] for all x; y 2 [ ; ] ; 2 C and < ( ) > 0: Proof. Using the Jensen -Mercer inequality, we have and so the …rst inequality of (2:1) proved. To be able to prove the second inequality in (2:1) , …rst we have to pay attention to that if f is a convex function, in case that, for 2 [0; 1] ; it gives Multiplying both sides of (2:8) by i : to both sides of (2:7) we …nd the second inequality of (2:1) : now we prove the inequality (2:2) :From the convexity of f we have Multiplying both sides of (2:11) by 1 e 1 and integrate with respect to over [0; 1] ; we have, The proof of …rst inequality of (2:2) is completed. On the other hand, using the convexity of f we can write by adding these inequalities an dusing the Jensen -Mercer inequality, we have Multiplying both sides of (2:15) by 1 e 1 and then integrating the resulting inequality with respect to over [0; 1] ; we have second and third inequalities of (2:2) : Remark 1. If we consider = 1; = 1 in Theorem;the following inequality is obtain Theorem 3. Let f be a positive continuous, decreasing and convex function on the interval [ ; ] then the following inequality for, generalized proportional fractional integrals holds; (2.16) Then where 2 (0; 1] for all x; y 2 [ ; ] ; 2 C and < ( ) > 0: Proof. To prove the …rst (2:16) ; by writting x 1 = 2 x + 2 2 y and y 1 = 2 2 x + 2 y for x; y 2 [ ; ] and 2 [0; 1] in the inequality (2:10) we get (2.17) 2f + and then, Multiplying both sides of (2:17) by 1 e 1 and integrate with respect to over [0; 1] ; we get the …rst inequality (2:16) is proved. To be able to prove the second inequality of (2:16) by using Jensen Mercer inequality , we get Multiplying both sides of (2:21) by

Hermite-Hadamard Mercer Type Inequalities For Fractional Integrals
Now, we give the new following lemmas for our results.
Lemma 1. Let f be a positive continuous, decreasing and convex function a di¤ erentiable mapping on ( ; ) with < : ; then the following equality for, generalized proportional fractional integrals holds; Proof. It is necessary to note, integrating by parts, we get (3.3) ( + We can write (3.5) i Multiplying the both sides by y x 2 , we proof is obtained (3:1) : Corollary 1. If we consider = 1; = 1 in Lemma 1; then we have the following equality