On generalized parameter-dependent Newton-type inequalities for proportional Caputo-hybrid operators

In this paper, we provide a thorough analysis of integral inequalities in the context of proportional Caputo-hybrid operators. Using these fractional operators, we first prove a new generalized parameter-dependent identity. We obtain a number of new error bounds for functions whose first and second derivatives are $ s $-convex in the second sense using this identity as an auxiliary result. Our main results are significant because they are general; we recover the classical Simpson's $ 3/8 $ rule, the corrected Simpson's $ 3/8 $ rule, and other related inequalities as special cases by appropriately adjusting the parameters $ \eta $ and $ \beta $. Furthermore, to demonstrate the validity and robustness of our theoretical findings, we provide numerical examples and graphical representations comparing the obtained bounds. These visualizations confirm the accuracy of the proposed estimates for varying fractional orders and convexity parameters.