Research Article | Open Access
Time Scale Inequalities of the Ostrowski Type for Functions Differentiable on the Coordinates
In 2016, some inequalities of the Ostrowski type for functions (of two variables) differentiable on the coordinates were established. In this paper, we extend these results to an arbitrary time scale by means of a parameter . The aforementioned results are regained for the case when the time scale . Besides extension, our results are employed to the continuous and discrete calculus to get some new inequalities in this direction.
To find a bound for the difference of a function and its integral mean, the Ukraine-born mathematician Ostrowski , in 1938, established the subsequent result which is nowadays celebrated as the Ostrowski inequality.
Theorem 1. Let be continuous on and differentiable in and its derivative is bounded in . If for all , then we havefor all The inequality is sharp in the sense that the constant cannot be replaced by a smaller one.
In 2001, Cheng  gave the following improvement of the above inequality.
Theorem 2. Let be continuous on and differentiable in such that there exist constants with for all Then for all , one gets
Theorem 3. Let , where , are open intervals in , be a mapping such that for , the partial mappingsdefined for all and , are differentiable, and Then we have
Theorem 5. Let , where are open intervals in , be a mapping such that for , the partial mappingsdefined for all and , are differentiable with , Then we have
In 1988, the idea of time scales  was initiated so as to bring together the continuous and discrete analysis into a unified fold. Since the introduction of this subject, many classical integral results have been extended to time scales. “A time scale is an arbitrary nonempty closed subset of .” We shall presume, all over this work, that the reader is familiar with the theory of time scale (see [5, 6] for more on this subject). We present here a result of Bohner and Matthews  which is embedded in Theorem 6 below. This result extends Theorem 1 to time scales. For more improvements and generalizations around this result, we refer the interested reader to see the papers [8–12] and the references therein.
Theorem 6. Let , and be differentiable. Then for all , we have where for all and This inequality is sharp in the sense that the right-hand side of (10) cannot be replaced by a smaller one.
It is our purpose in this paper to extend inequalities (4), (5), (6), (7), and (9) to time scales by means of a parameter . In Section 2, we frame and prove the main results followed by applications to the continuous and discrete calculus.
2. Main Results
Lemma 7 (see ). Let and be differentiable. Thenfor all such that and are in and , where This is sharp provided that
Lemma 8 (see ). Let and be differentiable. If and there exist such that for all , then for all , we havefor all such that and are in .
We now formulate and prove our first result.
Theorem 9. Let , with and be such that the partial mappingsdefined for all and , are differentiable. If and , then the succeeding inequalitieshold for all such that and .
Proof. Applying Lemma 7 to at , we getIntegrating (17) over givesApplying, again, Lemma 7 to at , and integrating over giveUsing (18) and (19), we haveSimilarly, doing the same thing for at and , and then integrating the resultant inequality over , we getUsing (20) and (21) amounts to (15). Also, from (20) and (21), we getCombining (22) and (23), one gets (16).
Theorem 10. Under the assumptions of Theorem 9 and suppose also the intervals contain the mid points, then we have
Proof. Next, we now apply Lemma 7 to at and thereafter integrate the resulting inequality over to getSimilarly, if one applies Lemma 7 to at and thereafter integrate the resulting inequality over , one getsCombining (26) and (27), we get (24). Finally, from (26) and (27), we getUsing (28) amounts to (25). Thus, the proof of Theorem 9 is complete.
Theorem 14. Let , with