symmetry

: The main goal of this work is to provide quantum parametrized Hermite-Hadamard like type integral inequalities for functions whose second quantum derivatives in absolute values follow different type of convexities. A new quantum integral identity is derived for twice quantum differentiable functions, which is used as a key element in our demonstrations along with several basic inequalities such as: power mean inequality, and Holder’s inequality. The symmetry of the Hermite-Hadamard type inequalities is stressed by the different types of convexities. Several special cases of the parameter are chosen to illustrate the investigated results. Four examples are presented


Introduction
The concept of convexity plays a significant role in the theory of inequalities. Inequalities have increasing importance in modern mathematical analysis, and in many other mathematical disciplines. Moreover, it seems to have "a pivotal role in several pure and applied sciences" [1]. Integral inequalities, on the other hand, "plays a vital role in the theory of differential equations" [2]. The very famous Hermite-Hadamard inequality is a part of integral inequalities and have been intensively studied by numerous scholars in the last decades. Different approaches have been followed in order to obtain new improvements, generalizations and refinements of this inequality [3][4][5][6][7][8][9] and of classical inequalities like Ostrowski, Simpson, Gruss, Chebysev, Mercer, Jensen, Hardy, Opial, Bullen, Newton, Bernoulli, Popoviciu and so on. Hermite-Hadamard's inequality(H-H inequality) and its different variants have been established for newly studied concepts as Riemann-Liouville fractional integrals [10], (k, s)-fractional integrals, fuzzy environment, quantum calculus, (p, q)-calculus and for different types of convexities. For example, a new type of convexity is the n-polynomial convexity investigated by Toplu in [11] where new refinements of Hermite-Hadamard were given. A possible reason for the great interest given to the study of the famous Hermite-Hadamard inequality may be the symmetry from within. The utilisation of the properties of modulus in the proof of all Hermite-Hadamard type inequalities involves a symmetry between the two expression obtained, the left member and the right member.
From a historical point of view, one can say that quantum calculus, as a branch of mathematics, was founded by Euler which used "the parameter q in Newton's work on infinite series" [12]. However, Jackson [13], started to develop the theory of quantum calculus when defined the notions of "general q-integral and q-difference operator" [12]. The q-fractional derivative was introduced firstly by Agarwal [14] and Al-Salam [10] defined the quantum analogue of the q-fractional integral and "q-Riemann-Liouville fractional integral" [12]. "The q-calculus concepts on finite intervals" [15,16] were utilized to obtain q-analogues of classical mathematical objects. "New quantum analogues of the Ostrowski inequalities" [17] have been described by Noor et al. and some estimated "bounds for the The case when the q-left and right derivatives of order three of the functions satisfies similar conditions was studied in [38] for convex functions. In all these inequalities we can see that in the expression from left member, the two integrals which appear are defined on different intervals, different from the corresponding intervals from inequalities [39,40] because of the parameter λ.
This approach could give interesting indications about how the variation of a quantity of the analyzed size varies(such as: utility, welfare economics, taxes, health or income inequalities) [41][42][43]. For the parameter, we choose λ = 1 and λ = 1 2 in our examples, which validating the theoretical results.
The paper has been structured in four sections. In Section 2, it will be briefly resume the basic notions and definitions of q-calculus. The classical H-H integral inequality is presented. Section 3 is dedicated to formulations and demonstrations of the main results: Lemma 2, Theorems 4-9. These theorems present new q-midpoints, trapezoidal and q-H-Hlike type integral inequalities for mappings whose the second q-left and q-right derivatives in absolute value satisfies different type of convexities (i.e., convex, strongly convex, npolynomial convex and strongly quasi-convex functions respectively). Many consequences have been established for some special choices of the parameter and the corresponding examples were discussed in detail. For the parameter we choose the values λ = 1 and λ = 1 2 in inequality from Theorem 5. We used for figures and several calculus the Matlab R2023a software. Section 4 is dedicated to discussion and conclusions.

Outcomes
Here, we recall some different types of convexities which will be used below in this paper. The classical convexity, say that a function Ψ : for all (x 1 , x 2 ) ∈ I 2 and t ∈ [0, 1]. Definition 1. Khan et al. [44] said that a function Ψ : holds for all (m, M) ∈ I 2 and τ ∈ [0, 1].

Definition 2.
Chu et al. [45] consider n ∈ N and a nonnegative function Ψ : I ⊂ R → R. This function is said then to be an n-polynomial convex function if for every (x, y) ∈ I 2 and t ∈ [0, 1], we have Definition 3. Kalsoom et al. [46] and Ion [47] said that a function Ψ : I → R with the modulus c is strongly quasi-convex function, if The well-known Hermite-Hadamard's inequality can be stated as "if Ψ : [a, b] → R is a convex function, then the following inequality holds: and when Ψ is a concave function, then previous inequality holds but in the opposite direction" [48]. This inequality is known also as trapezium inequality. Let suppose that [a, b] is a real interval with a < b. In this paper, it will be assumed that 0 < q < 1. It is well-known that the q-number is defined for any number n, [n] q = 1−q n 1−q = 1 + q + ... + q n−2 + q n−1 , n ∈ N.
Further, several basic definitions, remarks and lemmas of the q-calculus will be presented because they will be used throughout this paper.

Definition 7.
Alp et al. [50], and Rajkovic et al. [37] state the left or q a -integral of Ψ : For the fundamental properties of these q-derivatives and q-integrals, see for example, [16,51,52]. Recently, new refinements and generalizations of q-Hermite-Hadamard integral inequalities for q-differentiable functions were given in [37].
From now, we suppose that 0 < λ ≤ 1. By using Lemma 2 from [37] we will state again the following three theorems.

Theorem 1.
Alp et al. [37] show that if the conditions of Lemma 2 [37] hold and the | a D q Ψ| and | b D q Ψ| are convex on [a, b], then the following inequality holds: Theorem 2. Alp et al. [37] prove that if the conditions of Lemma 2 [37] hold and if the | a D q Ψ| s and | b D q Ψ| s , s > 1 are convex, then the following inequality holds: where s −1 + r −1 = 1.

Theorem 3.
Alp et al. [37] demonstrate that if the conditions of Lemma 2 [37] hold and if the | a D q θ| s and | b D q θ| s , s ≥ 1 are convex, then the following inequality holds: ].

Results
A new quantum identity with parameter is given below as an important instrument in the demonstrations of the results of this section. Some new estimates of parametrized q-Hermite-Hadamard-type integral inequalities for twice q-differentiable functions are given below having as a starting point the results formulated in [37]. Moreover, several new consequences, applications and examples are presented in order to check the established results. If Similarly, we have The main purpose of this paper is to give inequalities for b a S q (λ).

Lemma 2.
Let 0 < λ ≤ 1, and Ψ : [a, b] → R be a twice q-differentiable function. If, in addition, a D 2 q Ψ and b D 2 q Ψ are continuous and q-integrable functions over [a, b] then the following equality In addition, we can obtain also, b a S q (λ) = Proof. Denoting by I 1 the expression . From Definition 4, of the right q-derivative of Ψ, we successively have By Definition 6, of the right q-integral of Ψ and calculus, it will be obtained, Similarly, by using Definition 5, of the left q-derivative of Ψ, we successively get By Definition 7, of the left q-integral of Ψ and calculus, we have Then multiplying the result of the sum I 1 + I 2 by (b−a) 2 q 3 1+q , it follows: and we obtain the first desired equality.
For the second expression of b a S q (λ), we have, Similar we will obtain below the expression of I 2 Then multiplying the result of the sum I 1 + I 2 by (b−a) 2 q 3 1+q , we get the second expression of b a S q (λ), which completes the proof.

Remark 1.
First expression of b a S q (λ) has the coefficient 1 − q − q 2 which can take positive and also negative values, but the two integrals are defined on intervals which don't contain q. Also the second expression of b a S q (λ) is easier, but the integrals are defined on intervals more complicated which contain q.
In addition, expression of b a S q (λ) from Lemma 2, but especially in the second expression is preserving a symmetry of coefficients and terms. Theorem 4. It will be assumed that the hypothesis of Lemma 2 are true. If | a D 2 q Ψ| and | b D 2 q Ψ| are convex functions on [a, b] then we have: Proof. It will be used Lemma 2, obtaining: Then taking into account the convexity of | a D 2 q Ψ| and | b D 2 q Ψ|, we find that So by calculus it will be obtained the inequality from previous theorem.
Remark 2. Now considering λ = 1 in Theorem 4 the following trapezoid type inequality takes place: Remark 3. Now taking λ = 1 2 in Theorem 4 the following trapezoid type inequality holds: Remark 4. Now we put λ = 1 in Theorem 4 and we have: Theorem 5. We suppose that the hypothesiss of Lemma 2 takes place. If | a D 2 q Ψ| σ and | b D 2 q Ψ| σ are strongly convex functions on [a, b] for modulus c with σ ≥ 1, then next inequality is true: Proof. It will be used the modulus properties, the strong convexity of | a D 2 q Ψ| σ and | b D 2 q Ψ| σ and the power mean inequality, obtaining: and the proof is completed.

Theorem 6.
Under conditions of Lemma 2, if | a D 2 q Ψ| r and | b D 2 q Ψ| r are strongly convex functions on [a, b] with modulus c, when 1 s + 1 r = 1 and r > 1 then the following inequality takes place: Proof. By applying now, the properties of modulus, and the Holder's inequality, it will be obtained, and by using the strongly convexity of the functions | a D 2 q Ψ| r and | b D 2 q Ψ| r on [a, b] with modulus c, we get, Thus, the proof is finished.

Remark 5.
If we put λ = 1 in Theorem 5, then the following trapezoid type inequality holds: Remark 6. If we take λ = 1 2 in Theorem 5, then we obtain the following midpoint type inequality:

Remark 7.
If it is assigned in Theorem 6, λ = 1, then we have next trapezoid type inequality: Remark 8. If we take λ = 1 2 in Theorem 6, then the following midpoint type inequality it is obtained: Theorem 7. We assume that the conditions of Lemma 2 are true. If | a D 2 q Ψ| σ and | b D 2 q Ψ| σ are n-polynomial convex functions on [a, b], and σ ≥ 1, then we have: Proof. We can write as in the proof of Theorem 5 by using the power mean inequality and then the n-polynomial convexity of | a D 2 q Ψ| σ and | b D 2 q Ψ| σ , that Then following the calculus, we find that which leads to desired inequality.

Theorem 8.
We assume that the conditions of Lemma 2 are true and let s, r ∈ R, r > 1 with 1 s + 1 r = 1. If | a D 2 q Ψ| s and | b D 2 q Ψ| s are n-polynomial convex functions on [a, b], then we have: Proof. As in the proof of Theorem 6, by using Holder's inequality and then the n-polynomial convexity of | a D 2 q Ψ| s and | b D 2 q Ψ| s , we have,

}.
By calculus, evaluating the last four integrals involved in previous inequality, we find the desired inequality.

Theorem 9.
We suppose that all of the conditions of Lemma 2 are satisfied. If | a D 2 q Ψ| r and | b D 2 q Ψ| r are strongly quasi-convex convex functions on [a, b] for modulus c, where r > 1, with 1 s + 1 r = 1 then we have: Proof. We use the modulus properties, Holder's inequality and the definition of quasiconvex functions with modulus c for | a D 2 q Ψ| r and | b D 2 q Ψ| r , having An easy calculus shows that previous integral, , and the proof is completed.
where we put a = 0 and b = 1, obtaining, By calculus we obtain, For the right hand side of the inequality (3), we have where we put a = 0 and b = 1.
On the other hand, by calculus, . The graphic of the function 1 D 2 q Ψ(x) considered as function of two variables x and q, is given in Figure 1b. We see that the function is positive as 0 D 2 q Ψ(x). One can see the validity of the inequality (3) in Figure 1a, where the green line in graph represents the expression of the left member of the inequality (3) and the expression of the right member represents the magenta line in graph.
We used here the Matlab R2023a software for obtaining Figure 1 and also partial in calculus of last two derivatives.

Example 2.
For the same function Ψ : [0, 1] → R defined by Ψ(x) = x 4 , but with λ = 1 2 , which satisfies the conditions of Theorem 4, we obtain for the left hand side of the inequality (3), by similar calculus, the expression, ].
The right hand side of the inequality (3) becomes for λ = 1 2 , Like in Example 1, the red line in Figure 2 represents the left hand side of the inequality (3) from Theorem 4, and the blue line represents the right hand side of (3) and the inequality is checked.
where we put a = 0 and b = 1, obtaining, By calculus we obtain, For the right hand side of the inequality (3), we have where we put a = 0 and b = 1.
One can see the validity of the inequality (3) in Figure 3a, where the green line in graph represents the expression of the left member of the inequality (3) and the expression of the right member represents the magenta line in graph.

Discussion and Conclusions
The main findings of this study prove some new parametrized q-Hermite-Hadamard like type integral inequalities for functions whose second left and right q-derivative satisfies several different types of convexities. Some basic inequalities as q-Holder's integral inequality and q-power mean inequality have been used in order to obtain the new estimated bounds. An auxiliary q-lemma was utilized as a main tool in our proofs. Symmetry can offer an advantage in study of many processes and phenomena from nature. Interesting consequences arise for special choices of the parameter and the corresponding cases were discussed in detail.
We used the Matlab R2023a software for figures and for some calculus in the examples. Several consequences, examples and applications were given to illustrate the outcome of the research.
Furthermore, it is interesting to extend such findings to other new kinds of convexities, (p, q)-calculus, and q-fractional inequalities, which could be some good generalizations.
Even between the concept of convexity and the concept of symmetry there is a strong correlation, these two having many common ground to develop. This shows that the conclusions reached are pretty consistent. The study could be useful for the analysis of utility, distribution of taxes and revenues.
Overall, we hope that our results will improve the existing literature in the field.