Nonlinear Maligranda-Type and Dunkl-Williams Inequalities in Generalized Norm and Modular Structures

In this paper, we introduce a class of generalized nonlinear norm-like functionals obtained by adding an increasing nonlinear perturbation to a norm. The principal example is N(x)=‖x‖+‖x‖^2, which is positive, definite, continuous, but non-homogeneous and hence is not a norm. We establish a distorted triangle inequality and a nonlinear Maligranda-type reverse-triangle estimate with explicit magnitude-dependent coefficients. The sharpness of the principal coefficient is demonstrated, and the exact modulus of continuity of N on bounded balls is determined. We further obtain perturbation estimates for the general form N_φ (x)=‖x‖+φ(‖x‖), together with bounded-set Lipschitz and stability results. A nonlinear Dunkl-Williams-type inequality is also derived. Finally, the construction is extended to subadditive modular functionals, yielding corresponding modular-type inequalities.