On Some Generalized Simpson’s and Newton’s Inequalities for ( α , m ) -Convex Functions in q -Calculus

: In this paper, we ﬁrst establish two right-quantum integral equalities involving a right-quantum derivative and a parameter m ∈ [ 0,1 ] . Then, we prove modiﬁed versions of Simpson’s and Newton’s type inequalities using established equalities for right-quantum differentiable ( α , m ) convex functions. The newly developed inequalities are also proven to be expansions of comparable inequalities found in the literature.


Introduction
In [1], Hudzik and Maligranda introduced the concept of generalized convexity, called s-convexity and stated as follows: A function f : [0, ∞) → R is called s-convex or f ∈ K i s if the inequality f (tx + uy) ≤ t s f (x) + u s f (y) holds for all x, y ∈ [0, ∞), s ∈ (0, 1) and t, u ∈ [0, 1]. Note that if u s + t s = 1, the above class of convex functions is referred to as s-convex in the first sense and denoted by K 1 s , while if u + t = 1, the above class is referred to as s-convex in the second sense and denoted by K 2 s . Since much attention has been paid to investigating the idea of convexity and its variant forms in recent years, convexity concerning integral inequalities is an interesting research subject. Hermite inequality, Hadamard's Jensen's inequality, and Hardy's inequality are three of the most relevant inequalities linked to the integral mean of a convex function; see [2,3].
For a function to be convex, Hermite-Hadamard inequality is a necessary and sufficient condition. The Hermite-Hadamard inequality is given by: This double inequality is a development of the concept of convexity, and it readily follows from Jensen's inequality. Many researchers have investigated related Hermite-Hadamard type integral inequalities, and a remarkable diversity of generalizations and extensions for the concept of convexity have recently been considered.
In [4], Dragomir and Fitzpatrick used the s-convexity in the second sense for f and proved the following Hermite-Hadamard type inequality: primary findings. Sections 4 and 5 describe the Simpson's and Newton's type inequalities for q-differentiable functions via q-integrals. The findings provided here are compared to similar findings in the literature. Section 6 finishes with some research suggestions for the future.

Preliminaries of q-Calculus and Some Inequalities
The definitions and properties of quantum derivatives and quantum integrals are recalled first in this section. We also recall some well-known quantum integral inequalities. Throughout this paper, let 0 < q < 1 be a constant.
The q-number or q-analogue of n ∈ N is given by The function f is said to be q Note that, if a = 0 and 0 D q f (x) = D q f (x), then (3) reduces to which is the q-Jackson derivative; see [13,18] for more details.
Theorem 1. Ref. [18] If f , g : J → R are q-differentiable functions, then the following identities hold:  Note that, if a = 0, then (4) reduces to which is the q-Jackson integral; see [13,18] for more details.
Theorem 2. Ref. [18] If f : [a, b] → R is a continuous function and z ∈ [a, b], then the following identities hold: Alp et al. proved quantum Hermite-Hadamard inequalities for left q-integrals by utilizing the convex functions, as follows: Ref. [25] For a convex mapping f : [a, b] → R that is differentiable on [a, b], the following inequality holds: Definition 4. Ref. [19] The right q-derivative of mapping f : if it exists and it is finite.
Bermudo et al. also proved the corresponding quantum Hermite-Hadamard inequalities for the right q-integral: Theorem 4. Ref. [19] For a convex mapping f : [a, b] → R that is differentiable on [a, b], the following inequality holds: Now, we give another new lemma that helps us to prove the identities in the next section.

Lemma 1.
For continuous functions f , g : [a, b] → R, the following equality holds: Proof. The lemma can be shown by straightforward calculations, so it is omitted.

Identities
In this section, we establish two quantum integral equalities using the integration by parts method for quantum integrals to obtain the main results.
If D q f is continuous and integrable on [a, b], then one has the following identity for m ∈ [0, 1]: Proof. From the fundamental properties of quantum integrals, we have Using Lemma 1, we have Similarly, we have f (mb) (9) and Thus, from (8)-(10), we have and we obtain the required equality (6) by multiplying (mb − a) on both sides of (11). The proof is completed.
(iii) If we set α = m = 1 and later taking limit as q → 1 − , then we find Lemma 1 in [7].
We now observe how an equality emerges when we use the kernel mapping with three sections. on (a, b). If D q f is continuous and integrable on [a, b], then one has the following identity for m ∈ [0, 1]: Proof. The desired result can be attained if the same steps used in the proof of Lemma 2 are used in this proof.

Theorem 5.
Under the assumption of Lemma 2, if b D q f is (α, m)-convex mapping over [a, b], then we have the following Simpson-type inequality: where and Proof. By taking modulus in (6) and using the (α, m)-convexity of b D q f , we have Thus,the proof is completed.

Theorem 6.
Under the assumption of Lemma 2, if s ≥ 1 is a real number and b D q f s is (α, m)convex mapping over [a, b], then we have the following Simpson-type inequality: where Ω i (α; q), i = 1, 2, 3, 4 are defined in Theorem 5, and Proof. By taking the modulus in (6) and using the power mean inequality, we have . Now, applying (α, m)-convexity, we have Thus, the proof is completed.
(iii) If we set α = m = 1 and later taking limit as q → 1 − , then we find Theorem 7 for s = 1 in [7].

Theorem 7.
Under the assumption of Lemma 2, if s > 1 is a real number and b D q f s is (α, m)convex mapping over [a, b], then we have the following Simpson-type inequality: where s −1 + r −1 = 1.
Proof. Taking the modulus in Lemma 2 and applying Hölder's inequality, we have where one can easily observe that and, similarly, Thus, the proof is completed.

Theorem 9.
Under the assumption of Lemma 3, if s ≥ 1 is a real number and b D q f s is (α, m)convex mapping over [a, b], then we have the following Newton's type inequality: problem, and researchers can obtain similar inequalities for other kinds of convexity and coordinated (α, m)-convexity in future works.