On Weighted Montgomery Identity for Points and Its Associates on Time Scales
The purpose of this paper is to establish a weighted Montgomery identity for points and then use this identity to prove a new weighted Ostrowski type inequality. Our results boil down to the results of Liu and Ngô if we take the weight function to be the identity map. In addition, we also generalize an inequality of Ostrowski-Grüss type on time scales for points. For we recapture a result of Tuna and Daghan. Finally, we apply our results to the continuous, discrete, and quantum calculus to obtain more results in this direction.
In 1938, Ostrowski  proved the following inequality which approximates a function by its integral average.
Theorem 1. Let be a differentiable mapping on with the property that for all Thenfor all The constant is the best possible in the sense that it cannot be replaced by a smaller constant.
In 1997, Dragomir and Wang  obtained another inequality of this type.
Theorem 2. If is differentiable on and , for all for some constants , thenfor all
In 1988, the German Mathematician Hilger  introduced the concept of time scales. The time scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering formalism for studying hybrid discrete-continuous dynamical system. Since the introduction of this theory, it became a point of research to extend known classical differential and integral results to time scales. Following this line of thought, Bohner and Matthews  extended Theorem 1 to time scales by proving the following result.
Theorem 3. Let , , and be differentiable. Then, for all , one haswhere is given in Definition 12 and This inequality is sharp in the sense that the right-hand side of (3) cannot be replaced by a smaller one.
For more generalizations, extensions, and variants of Theorem 3, we refer the interested reader to papers [5–10] and the references therein. In 2008, Liu and Ngô  generalized Theorem 3 for points Specifically, they proved the following theorem.
Theorem 4. Suppose that (1) is a division of the interval for ,(2) is points so that and ,(3) is differentiable function. Then one haswhere This inequality is sharp in the sense that the right-hand side of (4) cannot be replaced by a smaller one.
Theorem 5. Let , and be differentiable. If is rd-continuous and , for all and for some , then, for all , one has
The aim of this paper is twofold, namely,(1)proving a generalized weighted Montgomery identity for points Using this identity, we then obtain a weighted version of Theorem 4. Our result boils down to Theorem 4, if the weight function is the identity map,(2)generalizing Theorem 5 for points For the case where , we recover Theorem 5.
This present paper is organized as follows. In Section 2, we provide some time scale essentials that will aid in better understanding of what follows. Our main results are then stated and proven in Section 3. Finally, we apply our results to the continuous, discrete, and quantum calculus to obtain more results in this direction. A brief conclusion follows thereafter in Section 4.
Definition 6. A time scale is an arbitrary nonempty closed subset of The forward jump operator and backward jump operator are defined by for and for , respectively. Clearly, we see that and for all If , then we say that is right-scattered, while if , then we say that is left-scattered. If , then is called right dense, and if , then is called left dense. Points that are both right dense and left dense are called dense. The set is defined as follows: if has a left-scattered maximum , then ; otherwise, For with , we define the interval in by Open intervals and half-open intervals are defined in the same manner.
Definition 7. The function , is called differentiable at , with delta derivative , if for any given there exists a neighborhood of such that
If , then , and if , then
Definition 8. The function is said to be -continuous if it is continuous at all right-dense points and its left-sided limits exist at all left-dense points
Definition 9. Let be -continuous function. Then is called an antiderivative of on if it is differentiable on and satisfies for any In this case, one has
Definition 10. The function is defined as for any
Theorem 11. If with , and are -continuous, then one has the following: (i)(ii).(iii).(iv)(v) .(vi)
Definition 12. Let , be functions that are recursively defined as
3. Main Results
For the proof of our theorems, we will need the following lemma.
Lemma 13 (generalized weighted Montgomery identity for points). Suppose that (1) is a division of the interval for ,(2) is points so that and ,(3) are differentiable functions. Then one has the following equation:where
Proof. Using items (i), (ii), (iv), and (vi) of Theorem 11, we have Hence, the result follows.
Remark 14. The above lemma becomes Lemma in paper  if we take
Corollary 15. If , then and (10) boils down towhere
Theorem 17 (weighted Ostrowski type inequality for points). Under the assumptions of Lemma 13, one has the following inequality:where
Remark 18. By choosing , we recapture Theorem 4.
Corollary 20 (discrete case). Let Suppose that (1) is a division of for ;(2) is points so that and ;(3) Then, for any differentiable function , and each , one haswhere and
Next, we formulate and prove a generalization of Theorem 5.
Theorem 21 (generalized Ostrowski-Grüss type inequality for points). Suppose that (1) is a division of the interval for ,(2) is points so that and ,(3) is differentiable, is rd-continuous, and there exist such that for all Then one has the following inequality:Inequality (20) is sharp in the sense that the constant 1/2 on the right-hand side cannot be replaced by a smaller one.
Proof. Using Lemma 13 with , we obtain (see also [11, Lemma ])whereAlso, from (22), we getSimilarly, one getsNow, let From assumption (), for all implies that for all This further implies that , for all Hence,Using (21) and (23), we obtainThe left-hand side of (26) is estimated as follows:Using relation (25) in (27), we getHence, the desired result follows.
We now apply Theorem 21 to different time scales.
Corollary 23 (continuous case). Taking in Theorem 21 amounts to the following inequality:
Proof. In this case, Using this in inequality (20) gives the desired result.
Corollary 25 (discrete case). Let Suppose that (1) is a division of for ;(2) is points so that and ;(3) Then one has the following inequality:
Proof. The proof follows by using the inequality in Theorem 21 and observing that
Corollary 26 (quantum case). Let with Suppose that (1) is a division of for ;(2) is points so that and ;(3) is differentiable. Then one has
Proof. Using Theorem 21 and the fact that, for the quantum calculus, one has ,
Corollary 27. Suppose that Then one has the following inequality:
The Ostrowski and Ostrowski-Grüss inequalities have received great deal of attention from the mathematical community dealing with inequalities. Giant steps have been made in extending some of the results to time scales. This work is tailored towards advancing this move. To be precise, we proved a generalization of the Montgomery identity and then used the resultant equation to obtain a weighted Ostrowski inequality for points, thus generalizing a result of Liu and Ngô . Furthermore, we obtained an Ostrowski-Grüss type inequality which generalizes and extends results of Tuna and Daghan  and Feng and Meng .
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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