Perturbed weighted trapezoid inequalities for convex functions with applications

We consider trapezoid type inequalities for twice differentiable convex functions, perturbed by a non-negative weight. Applications on a normed space \( (X, \lVert \,\cdot\, \rVert) \) are considered, by establishing bounds for the term \[ \begin{multline*} \frac{1}{2} \left[\lVert \frac{x+y}{2} \rVert^p + \frac{\lVert x \rVert^p + \lVert y \rVert^p}{2} \right] - \int_{0}^{1} \lVert (1-t)x + ty \rVert^p \, dt, \\ x, y \in X \end{multline*} \] which can be seen as a combination of both the midpoint and the trapezoid \(p\)-norm (with \(2\leq p<\infty\)) inequalities.