Estimation-type results on the k-fractional Simpson-type integral inequalities and applications

ABSTRACT We establish a Simpson-type identity of multiparameter and certain Simpson-type inequalities via k-fractional integrals. Worth mentioning, the obtained inequalities in this article generalize some results presented by Set et al. [Simpson type integral inequalities for convex functions via Riemann-Liouville integrals. Filomat. 2017;31(14):4415–4420] and Sarikaya et al. [On new inequalities of Simpson's type for s-convex functions. Comput Math Appl. 2010;60:2191–2199]. As applications, we also provide several inequalities for f-divergence measures and probability density functions. We expect that this study will be result in the new k-fractional integration explorations for Simpson-type inequalities.


Introduction
The following inequality is named the Simpson type integral inequality: where h : [τ 1 , τ 2 ] → R is a four-order differentiable mapping on (τ 1 , τ 2 ) and h (4) ∞ = sup t∈(τ 1 ,τ 2 ) |h (4) Considering the Simpson type inequalities, many researches generalized and extended them. For example, Hsu et al. [1], Du et al. [2], Noor et al. [3], İşcan et al. [4] and Tunç et al. [5] obtained some Simpson type inequalities for differentiable mappings which are convex, extended (s, m)-convex, geometrically relative convex, p-quasi-convex mappings and h-convex, respectively. Further results involving the Simpson type inequality in question with applications to Riemann-Liouville fractional integrals have been explored out by some scholars, including Set et al. [6] and Hwang et al. [7] in the study of the Simpson type inequalities using convexity, as well as İşcan [8] in the study of the Simpson type inequalities using s-convexity. More details corresponding to the Simpson type inequality and its extension, we refer to some articles by Hussain and Qaisar [9], Matłoka [10], Qaisar et al. [11], Ujević [12] and Ul-Haq et al. [13].
Let us consider an m-invex set A. A set A ⊆ R n is named m-invex set with respect to the mapping η : A × A × (0, 1] → R n for certain fixed m ∈ (0, 1], if mθ 1 + tη(θ 2 , θ 1 , m) holds, for all θ 1 , θ 2 ∈ A and t ∈ [0, 1]. A mapping h : A → R is called generalized (α, m)preinvex respecting η, if the following inequality holds, for every θ 1 , θ 2 ∈ A and t ∈ [0, 1]. Fractional calculus, as a very useful tool, shows its significance to implement differentiation and integration of real or complex number orders. This topic has attracted much attention from researchers who focus on the study of partial differential equations during the last few decades. For recent results related to this subject, we refer to some studies by Sohail et al. [14], Hameed et al. [15], and Khan et al. [16,17]. Among a lot of the fractional integral operators growed, the Riemann-Liouville fractional integral operator has been extensively studied, because of applications in many fields of sciences, such as differential equations, differential geometry and physics science. An important generalization of Riemann-Liouville fractional integrals was considered by Mubeen et al. in [18] which is named k-fractional integral operators. and respectively, where k > 0 and k (μ) is the k-gamma Some recent results pertaining k-fractional integrals can be found in [19][20][21].
Here, via k-fractional integral operators, we obtain some estimation-type results of Simpson-type inequality in terms of a multi-parameter identity. We also consider the established inequalities applying to f -divergence measures and probability density functions.

Main results
Throughout this article, let N * be the set of all positive integers, and let A ⊆ R be an open m-invex subset respecting η : We also utilize the following notation: h,η (μ, k; n, m) To prove main results, we give the following lemma.
then the following inequality with μ > 0 and k > 0 holds: Proof: Using Lemma 2.1, the Hölder integral inequality and the generalized (α, m)-preinvexity of |h (x)| ρ , we have Using the fact that From (15) and (16), we get the desired result in (14), since Proof: The second inequality is obtained by using the fact that n In the next theorem, we use the following functions.
For obtaining further estimation-type results, we next deal with the boundedness and the Lipschitzian condition of h .

Remark 2.4:
As several special cases of Theorems 2.3 and 2.4 above, some sub-results can be deduced by taking different mappings η and special parameter values for μ, k, n and m.

f-divergence measures
Let the set φ and the σ -finite measure μ be given, and let the set of all probability densities on μ to be defined on := {p|p : φ → R, p(x) > 0, φ p(x) dμ(x) = 1}. Let f : (0, ∞) → R be given mapping and consider D f (p, q) be defined by