FRACTIONAL SIMPSON’S MAJORIZATION INEQUALITY PERTAINING TWICE DIFFERENTIABLE FUNCTION WITH APPLICATIONS

Abstract

Over the past three decades, fractional calculus has gained increasing
importance and practical relevance in various fields of science and
engineering. This article aims to develop enhanced estimations based on the
Fractional Simpson’s rule for functions that are twice differentiable. Leveraging
majorization theory, we introduce a novel auxiliary identity by making use
of fractional integral operators. To derive the novel bounds presented in this
manuscript, we employ the notion of convex functions in conjunction with the
Niezgoda Jensen Mercer (JM) inequality for majorized tuples, as well as some
core inequalities, including Young’s, Power mean, and H¨older’s inequalities.
Furthermore, this study encompasses the application of quadrature rules and
provides illustrative examples related to special functions. Notably, the primary
contributions of this research involve the extension and generalization of
numerous well-established findings found in the current body of literature.