Mehmet Zeki Sarikaya

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.

In this paper, we develop a variance-based framework for the study of quadrature errors on arbitrary timescales. We introduce the timescale barycenter and the associated geometric variance, which allow the construction of a covariance operator adapted to the underlying timescale geometry. Using these notions, we establish a barycentric quadrature representation formula showing that the quadrature error can be expressed as a covariance between the independent variable and the delta derivative. As a consequence, we derive a fundamental variance inequality that bounds the barycentric quadrature error in terms of the geometric dispersion of the timescale and the variance of the delta derivative. The inequality is shown to be sharp, and a complete characterization of the equality case is obtained. Applications to the continuous, discrete, and quantum timescales are presented, illustrating the flexibility of the proposed framework and its ability to accommodate both homogeneous and non-homogeneous geometries.

In this paper, we investigate reverse Hardy-type inequalities associated with generalized weighted fractional kernel operators. The introduced operator framework unifies several important classes of integral operators, including classical Hardy operators, weighted Riemann–Liouville fractional integrals and generalized k-fractional operators. Using weighted kernel techniques, Hölder-type estimates and operator-theoretic methods, we establish new reverse Hardy inequalities involving mixed integrability parameters in weighted Lebesgue spaces. In addition, sufficient conditions for the boundedness of the corresponding fractional kernel operators are obtained. Several classical inequalities are recovered as particular cases of the developed framework. We also investigate the asymptotic behavior and near-optimality of the obtained constants by means of explicit power-type examples. Furthermore, applications to fractional integral equations and generalized weighted kernel operators are presented. The results obtained in this paper provide a unified approach to reverse Hardy inequalities for weighted fractional kernels and extend several existing results from the literature.

This paper establishes a series of novel Newton-type integral inequalities for twice-dierentiable mappings via the framework of Riemann-Liouville fractional integral operators. By leveraging the comprehensive properties of h-convex functions alongside a refined analytical approach, we derive generalized error bounds associated with the classical Simpson 3/8 quadrature rule. To enhance the versatility of the established boundaries, an explicit summation parameter p ≥ 1 is incorporated into our core estimates, thereby expanding the applicability of these inequalities across various general subclasses of convexity, including standard convex functions, s-convex functions, and P-functions. The results provided herein oer a unied and broader computational grid for error estimation in non-local numerical integration problems.

In this paper, we establish Lyapunov-type and Hartman–Wintner-type integral inequalities for boundary value problems involving a class of composite fractional integral operators. By employing an explicit Green function representation and sharp uniform bounds for the associated kernel, we derive necessary conditions for the existence of nontrivial solutions. A distinctive feature of the obtained results is the higher-order scaling behavior induced by the interaction of left- and right-sided components of the operator, which cannot be observed in classical one-sided fractional models. As applications, we obtain quantitative nonexistence criteria and explicit lower bounds for the principal eigenvalue of the corresponding eigenvalue problem. These estimates extend classical results to a higher-order fractional framework and highlight the influence of the domain geometry on the stability of the solutions.