Fractal-Fractional Perspectives of Trapezoid and Bullen Type Bounds With Applications

In this work, within the context of extended (fractional integral operators) (Formula presented.) in a fractal domain, we prove a new class of (trapezoid) (Formula presented.) and (Bullen) (Formula presented.) type inequalities that hold for differentiable convex functions over fractal sets. The primary advantage of utilizing these inequalities and their corresponding operators is their adaptability, which permits the transformation of inequalities on fractal sets inside the context of extended (Formula presented.). They also result in novel (fractal-fractional) F-F estimates and the corresponding optimized outputs. Additionally, we demonstrate the confirmation of these findings using a range of mathematical examples and the graphs that accompany them. To reinforce the significance of the findings, we additionally showcase the utilization of newly established findings concerning probability density functions, quadrature formulas, and special means.