Certain new bounds considering the weighted Simpson-like type inequality and applications

We investigate a weighted Simpson-type identity and obtain new estimation-type results related to the weighted Simpson-like type inequality for the first-order differentiable mappings. We also present some applications to f-divergence measures and to higher moments of continuous random variables.

Let us discuss several particular cases of Definition 1.1.
III. If h(t) = t, then Definition 1.1 reduces to the definition of (α, m)-convexity.
IV. If h(t) = 1, then Definition 1.1 reduces to the definition of (m, P)-convexity.  Also, the following theorem was proved in [19]. It obtains an estimation-type result associated with the weighted Simpson-type inequality for h-convex mappings using Hölder's inequality. Theorem 1.2 Let f : [r 1 , r 2 ] → R be a differentiable function on (r 1 , r 2 ) such that f ∈ L 1 [r 1 , r 2 ], and let w : [r 1 , r 2 ] → R be continuous and symmetric with respect to r 1 +r 2 2 . If |f | q is h-convex on [r 1 , r 2 ] for q > 1 and p -1 + q -1 = 1, then Different from [19] and [28], our purpose in this paper is to give some new bounds related to the weighted Simpson-like type inequality for the first-order differentiable mappings.
To obtain the principal results, we presume that the absolute value of the derivative of the considered mapping is (α, m, h)-convex. Next, we substitute this hypothesis with the boundedness of the derivative and with a Lipschitz condition for the derivative of the considered mapping to establish integral inequalities with new estimation-type results.
Also, we provide some applications to f -divergence measures and to higher moments of continuous random variables.

Main results
To obtain our main results, we need the following lemma.
and p 2 (t) = 1 4 Proof Integrating by parts and changing the variables, we have Similarly, we get Since w(x) is symmetric with respect to a+b 2 , we have Thus we have ba 4 (I 1 + I 2 ) which completes the proof.
Throughout the work, we write w [a,b],∞ = sup x∈ [a,b] |w(x)| for a continuous mapping w : [a, b] → R. Next, we derive our main results.
Proof Applying Lemma 2.1 and using the fact that Using the power mean inequality, we have Direct computation provides the following cases.

Corollary 2.2
If we take q = 1 in Theorem 2.2, then the following inequality for (α, m, h)convex functions holds: Remark 2.2 Consider Corollary 2.2.

Corollary 2.4 Consider Theorem 2.3.
(i) If we take h(t) = 1, then the following inequality for (m, P)-convex functions holds: (ii) If we take h(t) = t(1t) and α = 1, then the following inequality for (m, tgs)-convex functions holds: t and α = 1, then the following inequality for m-MT -convex functions holds: A similar result may be stated.

Theorem 2.4 Suppose that all assumptions of Theorem 2.3 are satisfied. Then
Proof The proof of Theorem 2.4 is analogous to that of Theorem 2.3 by using ta + (1t) a+b 2 = 1+t 2 a + 1-t 2 b and a+b The following result holds for (α, m, s)-convexity.
Remark 2.4 In Lemma 2.1, if we take w(x) = 1, then identity (2.1) becomes the following equation proved by Shuang et al. [28]: Remark 2.5 If we take h(t) = t and w(x) = 1 in Theorems 2.1 and 2.3, then we obtain Theorems 3.1 and 3.5 established by Shuang et al. [28], respectively.

Further estimation results
If the considered function f is bounded from below and above, then we have the following result.
where p 1 (t) and p 2 (t) are defined in Lemma 2.1.
Proof From Lemma 2.1 we have which implies that Also, since w is symmetric with respect to a+b 2 , we get This ends the proof.
Proof If we take w(x) = 1, then the relation w [a,b],∞ = 1 implies that Our next goal is an estimation-type result with respect to the weighted Simpson-like type inequality when the derivative of the considered function f satisfies a Lipschitz condition.
where p 1 (t) and p 2 (t) are defined in Lemma 2.1.
Proof From Lemma 2.1 we have Since f satisfies Lipschitz conditions for some L > 0, we have dμ(x) 1 2 . (4.5) Adding inequalities (4.4) and (4.5) and then using the triangle inequality, we get the desired result.

Random variable
Suppose that for 0 < a < b, w : [a, b] → [0, +∞) is a continuous probability density of a continuous random variable X that is symmetric about a+b 2 . Also, for r ∈ R, suppose that the rth moment Based on the above-mentioned derivations, we obtain the following estimates of the rth moment.
(a) If we consider f (x) = x r on [a, b] for r ≥ 2, then the function |f (x)| q = r q x q(r-1) with q > 1 is a convex function. Therefore, using this function in Remark 2.3 with s = 1 and in Corollary 2.5, respectively, we have E(X) r -E r (X)