Abstract

Inequalities become a hot topic for researcher due to its wide applications in means and sum, numerical integration, quantum calculus. Different generalizations and refinements are made by researchers. Here, in this article, we give another generalization of integral inequalities and harmonizing them on time scale from .

1. Introduction

Inequalities have a great contribution in mathematical analysis. In nonlinear analysis, these inequalities are very useful. Ostrowski’s inequalities have various coatings in numerical integration and in the theory of probability. In 1938, a mathematician A. Ostrowski gave an inequality named as Ostrowski inequality, since then a large number of results related to this inequality have been investigated by many researchers. In literature, many research papers appeared which contains refinements, elongations, generalizations and many similiar results of this inequality.

Theorem 1 (see [1]). Let be differentiable on , then we havewhere holds for all .

This is the Ostrowski inequality here the constant is best possible. Ostrowski’s inequality plays a vital role in theory of special means. This inequality has multiple ises in a variety of settings. Lately there have been elongations and many new results of this inequality. This inequality has significant and remarkable background in mathematical analysis. All the work related to this inequality is not possible to list here.

If you want to study discrete and continuous analysis together you will need the theory of time scale. S. Hilger competed the great task of harmonizing continuous and discrete calculus in one result, in his PhD research. Now we are able to give one definition for discrete and continuous analysis and if we change the range of function in the result we will come to different cases of time scale.

Time Scales is defined as a closed subset of by Stefan Hilger, which is symbolize as . A point of is defined as . If we consider then, . However, if then, , where and are forward and backward difference operators used in difference equability. The mappings defined as and are the jump operators. S. Hilger gave a new definition of derivative which was denoted by ; exists if and only if for every a neighborhood of s.t

Also a differentiable mapping is known as anti-derivative of on provided that , then

Let and ,

(1)If is differentiable at then is continuous at .(2)If is differentiable at , then

In the resent years, calculus of time scales has enchanted scientists due to its tremendous practical applications in many branches, e.g., quantum calculus, dynamical system, information theory, etc., see [2–4]. During the last decennia, the progression of integral and differential equation have been revealed. The convenient discoveries concern a consequential part in many areas of research of mathematics (can be seen in [5, 6]). S. Hilger has proposed the time scale theory in the terms “a theory that combines differential and difference calculus in the most worldly wise manner”. Concludingly, a number of researchers have discussed the new assorted fact of the dynamic inequations on time scales comprehensively [5, 7–11].

Lemma 2 (see [1]). Let . be differentiable, then , then.

where

Theorem 3 (see [1]). Let be differentiable, thenwhere .

This is sharp because the R.H.S of this inequality can’t be changed by any smaller number. In this paper, we also get a generalization of this inequality. In this article first of all we will prove a generalize form of montgomery identity and then discuss the case for In our next result get a generalized version of (7), we have also discussed its continuous, discrete and quantum calculus cases by choosing time scale as .

2. Main Results

For points The interval is distinguished as a real interval and is distinguished as . In this sense is a nonempty, closed and bounded set having points from . In this paper, by the interval we mean . Now we first prove an identity which is the generalize form of Montgomery identity and then use this identity in our next theorems to get new generalizations of Owstrowski’s inequality.

Lemma 4. Let , . If be differentiable, then , we havewherewith .

Proof. We initiated with

We can rewrite after calculations

Eventually, we come to the required result, i.e.,

Remark 5. Let then the above equation (8) becomeswhich is the Montgomery identity on talked in [9], also discussed in [1] with continuous, discrete and quantum cases.

Theorem 6. With supposition: for a time scale , such that . If be differentiable, then .

holds where .

Proof. We can rescript Lemma 4 asAnd further

Remark 7. When , then inequality (14) reduces towhich is the Ostrowski inequality on time scales as stated in (7).

Remark 8. Further choosing , respectively in inequality (14) with assumption of Theorem 6, we come to

Corollary 9 (Continuous Case). Let , then , and in the case -integral becomes usual Riemann integral, as Cauchy’s integral is a particular case of Riemann integral thus the inequality in (17) becomes

Corollary 10 (Discrete Case). Let then Also , thus the inequality in (17) becomes

Corollary 11 (Quantum Calculus Case). Let with In this situation we have therefore thus the inequality in (17) becomes

Theorem 12. Let . If be differentiable, and if is -continuous and ,holds wherewith .

Proof. Choosing in theorem 3.1 of [12], we have.

By solving and on , we get

that is

Thus, the R.H.S of inequality (25) becomes

From (25) and (27), we get

Now, from Lemma 4,

By using these inequalities, we come to the following result.

Remark 13. Let , then from (23), we have

Theorem 14. Let . If be differentiable, then for all ,

Proof. By using Lemma 4, we know that

where

Also

Let

On the other hand

we have also

therefore

Remark 15. Let , then.

Corollary 16 (Continuous Case). Let , then and in the case -integral becomes usual Riemann integral, thus the inequality

3. Concluding Remarks

The study of inequalities on is the genom of mathematics which is most recently gaining a substantial attention. The given article is the description of some general statements regarding Ostrowski’s type inequalities on . The results demonstrated here are some stimulus generalization of Ostrowski’s type inequalities via -integrals and generalizing the results of articles [8, 10–13]. These results will be very useful in the study of quantum calculus and dynamical system related differential equations which bring difference and differential equations together [14–18].

Data Availability

The authors confirm that the data supporting the findings of this article are available within the article and are available on request from the corresponding author.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

HEC, Pakistan is partially supporting in term of laboratory services, reference books, back volumes, Journals, stationery, software, Internet, computer etc.