Some generalized fractional integral Simpson’s type inequalities with applications

In the article, we establish a Simpson-type generalized identity containing multi-parameters and derive some new estimates for the generalized Simpson’s quadrature rule via the Raina fractional integrals. As applications, we provide several inequalities for the f -divergence measures and probability density functions.


Introduction
Over few years, the fractional calculus has attracted the attention of many researchers due to its has wide applications in pure and applied mathematics [1][2][3][4][5][6][7]. Like ordinary calculus, the fractional integral and derivative have not unique representation, with the passage of time, different authors have different representations. It is well-known that inequality is an indispensable research object in mathematics, it can give explicit error bounds for some known and some new quadrature formulae, for example, the Simpson's inequality [8], Jensen's inequality [9,10], Hermite-Hadamard's inequality [11][12][13][14][15] and integral inequalities [16][17][18][19][20][21]. The following inequality is well known as Simpson's inequality which provides an error bound for the Simpson's rule. Theorem 1.1. (See [7]) Let a, b ∈ R with a < b, and f : [a, b] → R be a four times differentiable function on (a, b) such that f (4) ∞ = sup x∈(a,b) | f (4) |(x) < ∞. Then the inequality holds.

Main results
To establish our results for generalized Simpson's type inequality using (s, p)-convex function, we need the following lemma.
Lemma 3.1. Let I ⊆ R + be an interval, I • be the interval of I, a, b ∈ I with a < b, ρ, β > 0, p ∈ R with p 0, and g(ξ) = p √ ξ for ξ > 0. Then the identity . Therefore, we get Again integrating by parts gives Then one has Therefore, the desired inequality (3.1) can be obtained by adding (3.2) and (3.3).
Proof. It follows from (3.1) and the (s, p)-convexity of | f | that
Proof. It follows from the (s, p)-convexity of | f | y and Hölder inequality that and Therefore, the desired inequality (3.9) follows from (3.5) and (3.10) together with (3.11).
Corollary 3.2. Let I ⊆ R + be an interval and I • be the interior of I, a, b ∈ I • with a < b, p < 0, u, v > 0, s ∈ (0, 1], g(ξ) = p √ ξ for ξ > 0, y = x x−1 > 1, and f : I → R be a differentiable function on I • such that | f | y is (s, p)-convex. Then the inequality holds.
Theorem 3.3. Let I ⊆ R + be an interval and I • be the interior of I, a, b ∈ I • with a < b, p, ρ, β > 0, x ≥ 1, u, w ∈ R, (s, λ) ∈ (0, 1] × [0, 1], g(ξ) = p √ ξ, and f : I → R be a differentiable function on I • such that | f | is (s, p)-convex. Then one has Proof. It follows from the (s, p)-convexity of | f | and the power-mean inequality that if f is convex, and the Hermite-Hadamard (HH) divergence D f Therefore, the desired inequality (4.3) can be derived by adding inequalities (4.4) and (4.5) to together with the triangular inequality.

Probability density functions
Let a, b ∈ R with a < b, g : [a, b] → [0, 1] be the probability density function of a continuous random variable X with the cumulative distribution function F given by (4.6) Then from Corollary 3.1 we clearly see that

Conclusions
We have established some new estimates for the generalized Simpson's quadrature rule via the Raina fractional integrals by use of a Simpson-type generalized identity with multi-parameters, and discovered several inequalities for the f -divergence measures and probability density functions. Our obtained results are the improvements and generalizations of some previous known results, our ideas and approach may lead to a lot of follow-up research.