On some fractional integral inequalities for generalized strongly modified h-convex functions

1 Basic Teaching Department, Shandong Huayu University of Technology, Dezhou, Shandong 253034, China 2 Department of Mathematics, Huzhou University, Huzhou 313000, P. R. China 3 Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, P. R. China 4 Department of Mathematics, University of Okara, Okara, Pakistan


Introduction
Linear functions are considered as simplest functions in linear spaces. The class of functions and sets that are just a step more complicated then linear ones namely convex functions and convex sets.
The subset C of R n is said to be convex if px + qy ∈ C ∀x, y ∈ C, p ∈ (0, 1) and q = 1 − p. The function f : R n → R is said to be convex if its epigraph is convex subset of R.The convexity of sets and functions are the the objects of many studies during the past few decades. The convexity of a function and set make it so special because of its interesting Definition 2.6. [22] A function f : ϕ = [b 1 , b 2 ] → R is called strongly convex function with modulus µ on ϕ, where µ ≥ 0 if for all b 1 , b 2 ∈ ϕ and t ∈ [0, 1].
Definition 2.8. [24] Choose the functions f, h : J ⊂ R → R are non-negative. Then f is called generalized strongly modified h-convex function if for all b 1 , b 2 ∈ J and t ∈ [0, 1].
[25] Let 0 < s ≤ 1. A function f : J ⊂ R → R is called s-φ -convex with respect to bifunction φ : The next remark provides the relations among the convexities.

Utilization of more complicated convex functions
Most of the modern problems in engineering and other applied sciences are non-convex in nature. So it is difficult to reach at favorite results by only the classical convexity. That's why the convexity is generalized in many directions. To understand the generalization of convexity it may categorize as: Some generalization are made to change the form of defining e.g. quasi convex [26], pseudo convex [27] and strongly convex [28]. Some generalizations are made by expanding the domain e.g. [29] and some generalization are made by changing the range set of convex functions e.g. [30]. So generalizations the convex is always appreciable.
The next lemmas are useful in proving the main results.
[23] If f n for n N exists and is integrable on (2.14) Lemma 2.14. [25] Suppose that f :

Fractional integral inequalities
, then using Lemma (2.10), and power mean inequality we have for q > 1 Using Lemma (2.11), we have And
Proof. From Lemma (2.12), we have Since | f | is generalized strongly modified h convex function, so This completes the proof.
Proof. Suppose that p≥ 1 , using Lemma (2.12) and power mean inequality, we have Since | f | q is generalized strongly modified h-convex, then we have After simplification, we have Which completes the proof.
Theorem 3.4. Let f :J ⊂ R → R be a n-times differentiable generalized strongly modified h-convex, ]. If f p is generalized strongly modified h-convex, function with µ ≥ 1, then for n ≥ 2 and p ≥ 1, we have (3.14) Proof. Case-i: Since it is known that f is generalized strongly modified h-convex function, then using the property of modules, and Lemma (2.13), we have following inequality for p = 1 Using the definition of generalized strongly modified h-convex function, we have and | f | is a generalized strongly modified h-convex function. Then where k = max t∈[0,1] |g(t)| and g(t) = h 1−t 2 + h 1+t 2 . Proof. From Lemma (2.14) and the fact that | f | is generalized strongly modified h-convex, we have where k = max t∈[0,1] |g(t)| and g(t) = h 1−t 2 + h 1+t 2 . Remark 9. If we take µ = 0, h(t) = t s then Theorem (3.15) reduces to Theorem (3) in [25]. (3.20) Corollary 5. In corollary (3.17) if we choose g = 1, η(x, y) = x − y, we have the following inequality for convex functions that is equivalent to Theorem (1.2) in [1].

Application to means
For two positive numbers b 1 > 0 and b 2 > 0, define Taking f (x) = lnx for x > 0 in Theorem (3.1) results in the following inequality for means.
Finally, we can establish an inequality for the Heronian mean as follows.

Conclusions
Fractional differential and integral equations play increasingly important roles in the modeling of engineering and science problems. It has been established fact that, in many situations, these models provide more suitable results than analogous models with integer derivatives. Fractional integral inequality results when 0 < q < 1 can be developed when the nonlinear term is increasing and satisfies a one sided Lipschitz condition. Using the integral inequality result and the computation of the solution of the linear fractional equation of variable coefficients, Gronwall inequality results can be established. In the present report, we developed the fractional integral inequalities for more broader class of convex functions named as generalized strongly modified h-convex functions, we also established some applications of derived inequalities to means. Our results extend and generalize many existing results, for example [1,23,[33][34][35].