Some inequalities obtained by fractional integrals of positive real orders

The primary objective of this study is to handle new generalized Hermite–Hadamard type inequalities with the help of the Katugampola fractional integral operator, which generalizes the Hadamard and Riemann–Liouville fractional integral operators into one system. In order to do this, a new fractional integral identity is obtained. Then, by using this identity, some inequalities for the class of functions whose derivatives in absolute values at certain powers are ρ-convex are derived. It is observed that the obtained inequalities are generalizations of some results in the literature.


Introduction
It is a well known fact that inequalities have important roles to play in the studies of linear programming, extremum problems, optimization, error estimates and game theory (see for example [2]). Over the years, only integer real order integrals were taken into account while handling new results about integral inequalities. However, in recent years, fractional calculus has been considered by many scientists (see [3-5, 7-10, 12, 13]). There are some inequalities in the literature that accelerated studies on integral inequalities. One of the most famous and practical inequalities in the literature was the Hermite-Hadamard inequality given in the following theorem. Theorem 1.1 Let f be defined from interval I (a nonempty subset of R) to R to be a convex function on I and a, b ∈ I with a < b. Then the double inequality given in the following holds: ( 1 ) Definition 1.1 Let I be an interval composed of positive real numbers and p ∈ R\{0}. f : I → R is called a p-convex function if it satisfies f tx p + (1t)y p 1 p ≤ tf (x) + (1t)f (y) ( 2 ) for all t ∈ [0, 1] and x, y ∈ I.
It is easy to see that ordinary convexity is retrieved from p-convexity for p = 1 and harmonically convexity is retrieved from p-convexity for p = -1. Now we will mention some kinds of fractional integral operators and the definition in the space X p c (a, b). The first of them is the Riemann-Liouville fractional integral, which makes the integration of fractional order possible (see [9]).
, which are called left-sided and right-sided Riemann-Liouville integrals of order α > 0 with a ≥ 0, are defined by and

Definition 1.3 ([9])
The left-sided and right-sided Hadamard fractional integrals of order α ∈ R α are defined as where Γ is the gamma function.
and, for the case p = ∞, Katugampola revealed a new fractional integration operator which generalizes both the Riemann-Liouville and the Hadamard fractional integration operators. This integration operator possesses the semigroup properties (see [4,5]) and is defined as follows. and with a < x < b and ρ > 0 if the integral exists. Equations (9) and (10) For right-sided operators, a similar conclusion can be drawn.
For more studies of fractional integral inequalities, see [10,12] and the references therein.
Erdelyi et al. were deeply involved in hypergeometric functions given in the following (see [1]): (13) and the regularized hypergeometric function is given in [11]. We will define T f (α, ρ; a, x, b) by and Γ is the Euler Gamma function, i.e., Γ (α) = ∞ 0 e -u u α-1 du. Kavurmacı et al. obtained new Ostrowski type results after proving the next lemma in 2011 in [6].

Lemma 1.1 ([6]) Let f be defined from an interval I to R as a differentiable mapping on the interior of I, where a, b ∈ I, a < b and f ∈ L[a, b]. Then the equality given here is valid:
Kavurmacı et al. presented the next lemma to handle Ostrowski type inequalities for Riemann-Liouville fractional integrals in 2012 in [7].

Lemma 1.2 ([7]) Let f be defined from interval I to R as a differentiable function on I • , where a and b belong to I with a < b and f ∈ L[a, b]. Then we get
for all x ∈ [a, b] and α > 0.
In this paper, a new kernel and Ostrowski type new theorems including the Katugampola fractional integral operator have been retrieved inspired by Lemma 1.2.

Lemma 2.1 Let f be defined from interval I which consists of positive real numbers to R as a differentiable function on I
Proof With the help of partial integration we have By changing the variable [tx ρ + (1t)a ρ ] 1 ρ = u we get Similarly we have By changing the variable [tx ρ + (1t)b ρ ] 1 ρ = u we get By multiplying (20) and (23) , respectively, and then summing them side by side, we have By rearranging the last equality we get the desired equality.
Proof Using Lemma 2.1 and the properties of the absolute value we get Then by taking into account the ρ-convexity of |f | and the Hölder inequality we get By the necessary computations we have where and for all x ∈ (a, b], α > 0, ρ > 0, r > 1, q > 1, 1 r + 1 q = 1, r = ρ ρ-1 .
Proof Using Lemma 2.1 and the properties of the absolute value we get By using the Hölder inequality we have Since |f | q is ρ-convex on I we get With simple calculation we get   ∈ (a, b)), α > 0, ρ > 1, q > 1, 1 ρ + 1 q = 1.
Proof Using Lemma 2.1 and the properties of the absolute value we get With the help of the power-mean inequality we have and by using the ρ-convexity of |f | q , then using the Hölder inequality we have By simple computation we get and K(a) = 1 0 tt α+1 tx ρ + (1t)a ρ 1 ρ -1 dt,