A Short note on the fractional trapezium type integral inequalities

: The authors have examined a large number of mathematical articles that deal with the expansion of the convexity technique and its various strategies, and focusing on it, they have determine the connection that can be developed among fractional trapezium type inequalities for generalized convex function. The authors advance this direction by examining the Hermite-Hadamard inequality by using h -preinvex functions.

The above inequality, Hermite-Hadamard inequality is one of the most important inequality for convex function.
This inequality has large applicability in the domain of stats and probability (Pecari'c, Proschan & Tong, 1992) [14] also include domain of functional analysis (Nicolescu & Peerson, 2006) [11]. In current years various analyst have investigated under the field of modern version in the advancement of the understanding of convex function. As represented in the following references ( [18]).
Further the extent progress of the method of convex function has been linked in the area of integral inequalities represent in the following paper(Vivas, Hernandez Hernandez & Merentes, 2016) [16], (Vivas-Cortez, 2016) [17].
Influenced by the valuable work stated above, we modified the following work by examining the Hermite-Hadamard inequality using h-preinvex function.

Preliminaries
We identify the following definition linked with the h pre-invex .
If the given inequality true for the set k ⊆ R n and u, v ∈ K then a function f is known as h pre-invex where η(u, v) : k * k → R and s ∈ (0, 1) and h = 0 be a non negative function h : [0, 1] → R .

Remark 2.1.
We obtain definition of classical convex function by putting h(s)=s and η(v, u) = v − u in definition 1.

Definition 2.3.
Let [c, d] ⊂ R be a finite interval. The katugampola fractional integrals of order α > 0 of f ∈ X ρ c (a, b) on the leftand right-sides are therefore denoted by

Remark 2.2.
Let α > 0 and ρ > 0. Then for u > c lim The same is true for right-handed operators.

Definition 2.4.
Let k ⊆ R n be the set and u, v ∈ K then a function f is said to be (ψ, h) pre-invex if where η(u, v) : k * k → R and s ∈ (0, 1) and h = 0 be a non negative function h : [0, 1] → R .
, then the given inequality true:
is also convex. The function F has several interesting properties, especially, Theorem 3.1.
If f is a convex function on [c, c + η(d, c)] and f ∈ L[c, c + η(d, c)]. Then F (x) is also integrable, and the following inequalities hold with α > 0 and ρ > 0. Proof Using the notation of F (x), we have Multiplying both side of (3.3) by integrating the resulting inequality with respect to s over [0,1], we get Similarly multiplying both sides of (3.3) by integrating the resulting inequality over [0,1], we get By adding (3.5) and (3.7), we obtain The first inequality of (3.2) is proved.
For the second part, since f is convex function, then for t ∈ [0, 1], we have Using the notation of F (x), we then have Multiplying both sides of (3.9) by factor (3.4) and integrating the resulting inequality over [0,1], with respect to Similarly multiplying both sides of (3.9) by factor (3.6) and integrating the resulting inequality over [0,1], with respect to s, we get Adding (3.10) and (3.11), we get The proof is complete.
In order to prove Theorem 3, we need the following lemma.
We are now ready to prove the following Hermite-Hadamard type inequality.

Concluding Remarks
Here we determine major results related to Hermite-Hadamard inequality using h pre-invexity. A couple of results in the previous research paper are special cases of few of our results. The modified integral inequalities give a more precise approximation than some of the preceding ones.