Generalized Two-Sided Ostrowski-Type Inequalities on Time Scales Under Variable Derivative Bounds

This paper establishes a family of two-sided Ostrowski-type inequalities on arbitrary time scales under variable derivative bounds. The main approach is based on auxiliary functions that provide variable lower and upper envelopes for the delta derivative, thereby extending the usual framework based on constant derivative bounds. Starting from a generalized Montgomery-type identity, we derive two-sided estimates for the deviation of a function from its delta-integral mean. Further refinements are obtained under suitable monotonicity and convexity assumptions, together with Hölder-type estimates. The general formulation recovers continuous, discrete, uniform lattice, and quantum cases as particular realizations of the time scale framework. In addition, the obtained estimates are applied to midpoint and trapezoidal quadrature formulas, yielding explicit error bounds. A numerical example is included to illustrate the resulting estimates.