A Unified Representation of q - and h -Integrals and Consequences in Inequalities

: This paper aims to unify q -derivative/ q -integrals and h -derivative/ h -integrals into a single definition, called q − h -derivative/ q − h -integral. These notions are further extended on the finite interval [ a , b ] in the form of left and right q − h -derivatives and q − h -integrals. Some inequalities for q − h -integrals are studied and directly connected with well known results in diverse fields of science and engineering. The theory based on q -derivatives/ q -integrals and h -derivatives/ h -integrals can be unified using the q − h -derivative/ q − h -integral concept.


Introduction
The subject of q-calculus is based on the quotient f (qx)− f (x) (q−1)x involved in the derivative of a function.This motivates researchers to consider whether the results and theory that hold for usual derivatives can be further developed by analyzing this quotient.Euler (1707-1783) was the first to work in this direction, introducing the number q in the infinite series defined by Newton.Jackson [1,2] continued the work of Euler and defined q-derivatives and q-integrals.Roughly speaking, q-calculus analyzes q-analogues of mathematical concepts and formulas that can be recaptured by the limit q → 1.The concepts of q-calculus are extensively applied in various subjects of physics and mathematics, including combinatorics, number theory, orthogonal polynomials, geometric function theory, quantum theory and mechanics, and the theory of relativity; see [3][4][5][6][7][8][9].
Nowadays, many authors are applying quantum calculus theory in their fields of research.Consequently, they have contributed plenty of articles in this active field.For instance, theory of fractional calculus, optimal control problems, q-difference, and q-integral equations are studied in q-analysis; see [10][11][12][13] and references therein.In [14,15], Tariboon et al. defined quantum calculus on finite intervals and extended some important integral inequalities using this concept.
Here, our goal is to unify quantum calculus (q-calculus) and plank calculus (h-calculus).For this purpose, the notion of q − h-derivatives is introduced and basic calculus formulas are presented.Moreover, a q − h-binomial is constructed and the q − h-integral is defined on a finite interval.Using the q − h-integral, Hermite-Hadamard type inequalities can be constructed which combine the inequalities for qand h-integrals in implicit form.By imposing the symmetric condition, a correct proof of an already published inequality (the first inequality in Equation ( 12)) is provided.Next, we begin to lay out well known initial concepts which are useful for the reader in understanding the findings of this paper.
The h-derivative and the q-derivative of a function ν are defined by the quotients respectively.The h-derivative is usually denoted by D h ν(γ) = d h ν(γ) d h γ and the q-derivative is denoted by D q ν(γ) = d q ν(γ) for the function ν.As an example, the h-derivative and the q-derivative of γ n can be computed in the forms , respectively.For the sake of simplicity, the notation [n] q is used instead of q n −1 q−1 ; thus, dγ , the h-derivative and the q-derivative are generalizations of ordinary derivative.The q-derivative leads to the subject of q-calculus; see [16] for details.
The sum and product formula of q-derivatives for functions ν 1 and ν 2 are provided by and 2) is equivalent to the formula In view of Equation ( 2), the quotient formula of q-derivatives is provided by In view of Equation ( 3), the quotient formula of q-derivatives is provided by The formulae for the h-derivatives are as follows: and Next, we provide the definition of a q-derivative on a finite interval.

Definition 1 ([15]
).Let µ : I = [a, b] → R be a continuous function.For 0 < q < 1, the q-derivative a D q µ on I is provided by Function µ is called q-differentiable on [a, b] if a D q µ(ξ) exists for all ξ ∈ [a, b].For a = 0, we have 0 D q µ(ξ) = D q µ(ξ); moreover, D q µ(ξ) is the q-derivative of µ at ξ ∈ [a, b], defined as follows: The q-integral of the function µ on interval [a, b] is defined below.
Definition 2 ([15]).Let µ : I = [a, b] → R be a function.For 0 < q < 1, the q-definite integral on I is provided by In the following we provide a q-integral inequality published in [15].
Theorem 1 ([15]).Let µ : [a, b] → R be a convex continuous function on [a, b] and let 0 < q < 1; then, we have In (11), setting a = 0, the Jackson q-definite integral in [16] is deduced as follows: If c ∈ (a, ξ), then the q-definite integral on [c, ξ] is calculated as follows: We intend to unify the q-derivative and h-derivative into a single notion, which we name the q − h-derivative.We provide sum/difference, product, and quotient formulas for q − h-derivatives, along with the definition of the q − h-integral.Further, we define the q − h-derivative and q − h-integral on a finite interval.The composite derivatives and integrals provide the opportunity to simultaneously study theoretical and practical concepts and problems from different fields related to q-derivatives and h-derivatives.For instance, in Theorem 3 we prove the generalization of the inequality in (12) via the q − h-integral.

Generalization of q-and h-Derivatives
We define the (q − h)-differential of a real valued function µ as follows: Then, for h = 0 and q → 1 in (15), we have and Then, for h = 0 and q → 1 in ( 16), we have For u(ξ) = µ(ξ) + ν(ξ), the (q − h)-differential of u is provided by For α ∈ R, the (q − h)-differential of αµ is provided by From ( 18) and ( 19), it can be seen that the (q − h)-differential is linear.If p(ξ) = µ(ξ)ν(ξ), then the (q − h)-differential is calculated as follows: Hence, we obtain For h = 0 and q → 1 in (20), we have and respectively.Next, we define the q − h-derivative as follows: Definition 3. Let 0 < q < 1 and h ∈ R, and let µ : I → R be a continuous function.Then, the q − h-derivative of µ is defined by provided that q(ξ + h) ∈ I.
For h = 0 and q → 1 in (21), we have and By setting h = 0, q → 1 in (21), we obtain the ordinary derivative of µ, provided that the limit exists.
For h = 0 and q → 1 in (24), we have and In particular, we have lim

Linearity
The q − h-derivative is linear, i.e., for α, β ∈ R and using the linearity of (q − h)-differentials, we have

Product Formula
The following formula for a product of functions can be obtained using (20): The product formula for q-derivatives and h-derivatives can be obtained as follows.
By setting h = 0 in (27), the following q-derivative formula for products of functions is yielded: By taking q → 1 in (27), the following h-derivative formula for products of functions is yielded: Using symmetry, from (27) we have the following: Both ( 27) and (30) are equivalent.

q − h-Derivative on a Finite Interval
Throughout this section, I := [a, b] for a, b ∈ R. The q − h−derivative on I is provided in the upcoming definition.Definition 4. Let 0 < q < 1, h ∈ R, and ξ ∈ I, and let µ : I → R be a continuous function.Then, the left q − h−derivative C h D a + q µ and right q − h−derivative C h D b − q µ on I are defined by We say that µ is left q − h-differentiable on (a, x + h) if C h D a + q µ(ξ) exists for each of its points, and we say that . In (39), by setting h = 0, it is possible to obtain the q-derivative defined in Definition 1, i.e., C 0 D a + q µ(ξ) = a D q µ(ξ).Similarly, for a = 0 we can have C h D 0 + q µ(ξ) = C h D q µ(ξ), i.e., the q − h-derivative in (21) is deduced; for h = 0 = a, we can have C 0 D 0 + q µ(ξ) = D q µ(ξ), i.e., the q-derivative is deduced; for a = 0, q = 1, we can have C h D 0 + 1 µ(ξ) = D h µ(ξ), i.e., the h-derivative is deduced; and for h = 0 = a, taking the limit q → 1, we can obtain the usual derivative for a differentiable function µ, i.e., lim It is possible to obtain similar results from Equation (40).The definition of left and right q−derivatives defined on I can be obtained from (40) by setting h = 0, as follows.
Definition 5. Let 0 < q < 1, h ∈ R, and ξ ∈ I, and let µ : I → R be a continuous function.Then, the left q−derivative D a + q µ and right q−derivative D b − q µ on I are defined as follows: It is notable that from (41) we have D 0 + q µ(ξ) = D q µ(ξ), i.e., the left q−derivative coincides with the q−derivative defined in Definition 1. Definition 6.Let 0 < q < 1 and µ : I = [a, b] → R be a continuous function.Then, the left q − h-integral I a+ q,h µ and right q − h-integral I b q−h µ on I are defined as follows: The left and right q − h-integrals are calculated as follows: and where 1 is the radius of convergence of the series involved in the above integrals.
Example 3. Let µ(γ) = ξ − γ and ν(γ) = γ − ξ; then, we have and where 1 is the radius of convergence of the series involved in the above integrals.
By setting h = 0, the left and right q-integrals can be obtained and defined as follows.
Definition 7. Let 0 < q < 1 and µ : I = [a, b] → R be a continuous function.Then, the left q-integral I a+ q µ and right q-integral I b q µ on I are provided by The left q-integral is the same as the q a -definite integral defined in [15], while the right q-integral is the same as the q b -definite integral defined in [18]. .
By considering q → 1, we can include the left and right h-integrals in the upcoming definition.
Note that from Definition 6 we have

Some q − h-Integral Inequalities for Convex Functions
In this section, we provide inequalities for q − h-integrals of convex functions.A function µ : [a, b] → R is called convex if the following inequality holds for all u, v ∈ [a, b] and λ ∈ [0, 1]: Theorem 2. Let µ : J → R be a convex function and let a, b ∈ J • (the interior of J).The left and right q − h-integrals satisfy the following inequalities: + µ(ξ) q(ξ − a) 1 + q + (1 − q)hS and , we obtain the following inequality: By taking the q − h-integral over [a, ξ], we have Using the values of the integrals involved in the above inequality from (45) and (47), we can obtain the required inequality (54).On the other hand, for γ ∈ , we obtain the following inequality:

By taking the
Using the values of the integrals involved in the above inequality from ( 46) and (48), we can obtain the required inequality (55).

Corollary 1.
As an application of the above theorem, the following inequalities for left and right q-integrals hold: and Remark 2. By taking ξ = b in (56) or ξ = a in (57), we can obtain the following inequality: The above inequality (58) is independently proved in ( [15], Theorem 12).
The following lemma is required to prove the next result.
Proof.A convex function that is symmetric about a+b 2 satisfies the inequality in (59); therefore, by taking q − h-integration of (59) over [a, ξ] we have On the other hand, by taking the q − h-integration of (59) over [ξ, b], we have By adding (61) and (62), we obtain the inequality in (60).
Remark 3. By taking x = b in (61) or x = a along with h = 0 in (62), we can obtain the following inequality: The above inequality (63) is independently proved in ([15] Theorem 3.2).Unfortunately, the proof is not correct; see ( [20] Example 5).Here, we have imposed an additional symmetric function condition to ensure the result.Hence, if we impose a condition of symmetry in addition to the assumptions in ([15] Theorem 3.2), we obtain the correct result.

Conclusions
This article provides a base for unifying the theory of q− and h-derivatives provided in [16] by Kac and Cheung.The notion of a q − h-derivative that generates the q-derivative and h-derivative is introduced.The q − h−binomial (ξ − a) n h,q analogue to (ξ − a) n is defined, which generates the q−binomial (ξ − a) n q and h−binomial (ξ − a) n h in particular.The q − h-derivatives of the q − h−binomial (ξ − a) n h,q are found, which generate the qderivative of the q−binomial (ξ − a) n q and h-derivative of the h−binomial (ξ − a) n h in particular.The rest of the theory in [16] needs further attention from researchers, as it may be unified in a similar way to the q − h-derivative and q − h−binomial.In addition, the q − h-derivatives and integrals are defined on an interval [a, b], which is used to establish some inequalities linked to recent research and provide a corrected proof of the inequality in [15]."The composite derivatives and integrals create the opportunity to study theoretical and practical concepts and problems of different fields related to q-derivative and h-derivative simultaneously".

Definition 8 .
Let µ : I = [a, b] → R be a continuous function; then, the left h-integral I a+ h µ and right h-integral I b h µ on I are defined as follows: