New Generalized Ostrowski-Grüss Type Inequalities In Two Independent Variables On Time Scales

Recently, the research for the Ostrowski type and Grüss type inequalities has been paid much attention by many authors. The Ostrowski type inequality, which was originally presented by Ostrowski in [1], can be used to estimate the absolute deviation of a function from its integral mean, while the Grüss inequality [2] can be used to estimate the absolute deviation of the integral of the product of two functions from the product of their respective integrals. Among the research for the Ostrowski type and Grüss type inequalities, generalizations of the two inequalities have been a hot topic, and, in the last few decades, various generalizations of the Ostrowski inequality and the Grüss inequality have been established (for example, see [3-13] and the references therein), while some new inequalities are established, one of which is the inequalities of Ostrowski-Grüss type (for example, see [14-26]). The first Ostrowski-Grüss type inequality was presented by Dragomir and Wang in [14], which got improved in [15-18, 21-23]. In [25], Lü extended UJEVIĆ’s results [17] to 2D case. In [26], Liu extended the inequality above to a more general form. The most important application for these inequalities mentioned above lies in that they can be used to provide explicit error bounds for some known and some new numerical quadrature formulae, and furthermore can provide sharp bounds related to these inequalities. So establishing newOstrowski type and Grüss type inequalities is a purposeful work in estimating new error bounds for numerical quadrature formulae. On the other hand, Hilger [27] initiated the theory of time scales as a theory capable of treating continuous and discrete analysis in a consistent way, based on which some authors have studied the Ostrowski type and Grüss type inequalities on time scales (see [28-38]). But we notice that Ostrowski-Grüss type inequalities in two independent variables on time scales have been paid little attention in the literature. Motivated by the above works, in this paper, we establish some new Ostrowski-Grüss type inequalities in two independent variables on time scales with more generalized forms than those existing inequalities in the literature. New bounds related to the OstrowskiGrüss type inequalities are derived, and some of them are sharp. The established results unify continuous and discrete analysis, and extend some known results in the literature. Throughout this paper, R denotes the set of real numbers, while Z denotes the set of integers, and N0 denotes the set of nonnegative integers. T1, T2 denote two arbitrary time scales, and for an interval [a, b], [a, b]Ti := [a, b] ∩ Ti, i = 1, 2. For the sake of convenience, we denote the forward jump operators on T1, T2 by σ uniformly. Finally, a point t ∈ Ti is said to be right-dense if σ(t) = t and t ̸= supTi. For more details about the calculus of time scales, we refer the reader to [39-40].