Extensions of Ostrowski Type Inequalities via h-Integrals and s-Convexity

In mathematics, the quantum calculus is equivalent to usual infinitesimal calculus without depending upon the concept of limit. It has two major branches, q-calculus and the h-calculus. It is really the calculus of finite differences, but a more systematic analogy with classical calculus makes it additionally transparent. (e definite h-integral is a Riemann sum so that the fundamental theorem of h-calculus allows one to evaluate finite sums and h-integration by parts which is simply the Abel transform. (e theory of h-discrete calculus is the rapidly developing area of great interest both from theoretical and applied point of view. (is calculus is the study of the definitions, properties, and applications of the related concepts, the fractional calculus and discrete fractional calculus.


Introduction
In mathematics, the quantum calculus is equivalent to usual infinitesimal calculus without depending upon the concept of limit. It has two major branches, q-calculus and the h-calculus. It is really the calculus of finite differences, but a more systematic analogy with classical calculus makes it additionally transparent. e definite h-integral is a Riemann sum so that the fundamental theorem of h-calculus allows one to evaluate finite sums and h-integration by parts which is simply the Abel transform. e theory of h-discrete calculus is the rapidly developing area of great interest both from theoretical and applied point of view. is calculus is the study of the definitions, properties, and applications of the related concepts, the fractional calculus and discrete fractional calculus. [1].

Definition of h-Integral
With this definition, the definite h-integral is Riemann sum of ϕ(θ) on the interval [], μ], which is proportioned to subintervals of equal width. [1]. Let ϕ, g: [], μ] ⟶ R be the continuous functions and θ ∈ [], μ], then the formula of h-integration by parts is stated as

Properties of h-Calculus
e h-analogue of a binomial expansion (θ − ]) n is defined as Note that h-analogues of an integer n is still n, and (θ − 0) n h ≠ θ n . [2]. Let t, α ∈ R, then the h-fractional function t (α) h is defined by

h-Fractional Function
where Γ is the Euler Basic inequalities have a massive role both in pure and applied sciences in the light of their wide applications in mathematics and physical sciences, while convexity theory has stayed as a significant instrument in the foundation of the hypothesis of integral inequalities. e following Ostrowski inequality [3] is notable to read.
In [25], the class of functions which are called s-convex in the second sense has been introduced by Hudzik and Maligranda as follows: for each θ, ϕ ∈ R + , Ω ∈ [0, 1] and for unique s ∈ (0, 1]. e integral equality is established by Alomari et al. in [26].

Main Results
Initially, we establish the following identity.
, then the following h-integral equality is valid: for each θ ∈ [], μ]. Proof. By formula (4) of h-integration by parts, the first term of right hand side of (15) becomes and the second term of right hand side of (15) becomes From (16) and (17), □ Journal of Mathematics for each θ ∈ [], μ].
Proof. From Lemma 2 and keeping the familiar Hölder inequality, we have Proof. From Lemma 2 and keeping in view the familiar power mean inequality, we get Journal of Mathematics Example 1. In case of time scale ⊤ � Z in Lemma 2, we have By taking h � 1, in eorem 5, we have By taking h � 1, in eorem 6, we have By taking h � 1, in eorem 7, we have Journal of Mathematics By taking h � 1 and s � 1 in eorem 5, we have By taking h � 1 and s � 1 in eorem 6, we have By taking h � 1 and s � 1 in eorem 7, we have

Conclusion
Our results extend and generalize the results of Alomari et al.
In this work, some important Ostrowski type inequalities are established in the context of h-calculus. e derived results constitute contributions to the theory of h-integral and can be specialized to yield numerous interesting integral inequalities including some known results. An interesting feature of our results is that they provide new estimates and best approximation on Ostrowski type of inequalities for h-integral. If we take limit h ⟶ 0, Ostrowski type of hintegral inequalities reduces to simple inequalities present in [26]. e presented results motivate scientists to stimulate more work in such directions.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.