Comprehensive Weighted Newton Inequalities for Broad Function Classes via Generalized Proportional Fractional Operators

In this work, weighted Newton-type inequalities (WN-TI) for multiple classes of functions by employing generalized proportional fractional (GPF) integrals are established. The core step involves establishing a key integral identity under a positive weighting scheme, which underlies all subsequent results. By integrating this identity with GPF operators, we derive various forms of (WN-TI) suitable for several function types, including differentiable convex, bounded, and Lipschitz functions.