In this manuscript, firstly we introduce a new auxiliary result for the Caputo-Hybrid fractional operator $(P_{cap})$ . As a result, we obtain improved hybrid estimates of Euler-Maclaurin-type for both first- and -second-order differentiable convex functions within the fractional calculus context. The $(P_{cap})$ operator offers a versatile and unified framework to study derivatives: when the fractional order α=1, it simplifies to the standard first-order derivative; for $\alpha =0$ , it approximates the second-order derivative and at $\alpha =\frac{1}{2}$ , it gives a hybrid form that combines the properties of both first- and -second-order derivatives. Various important inequalities are derived through convexity conditions, combined with traditional tools like Hölder’s and Power mean inequality. In addition, we further explore new fractional inequalities for bounded and Lipschitzian functions, highlighting the usefulness of the $({P_{cap}})$ operator across different functional contexts. Representative examples and graphical illustrations are given to confirm the sharpness and efficiency of the main conclusions. Special cases are further explored to exhibit the wider utility of the proved theorems. As additional contributions, novel inequalities are derived for modified Bessel functions of the kind-1, special means and Maclaurin quadrature formulas related to the $({P_{cap}})$ operator.
