Stability of h–convex functions and applications to integral inequalities

We develop a two-regime stability and transform framework for h –convex functions. The notion of ε – h –convexity is introduced, and both a contractive regime and a normalized noncontractive regime are studied. In the contractive case, assuming that aligned h(α )+h(1-α )≤ L<1, aligned we prove a collapse-type Hyers–Ulam stability theorem. More precisely, every nonnegative ε – h –convex function is uniformly controlled by the zero h –convex envelope, and the sharp estimate aligned x∈ I|f(x)-g(x)|≤ ε /1-L aligned is obtained. In this regime, the exact nonnegative h –convex class reduces to the zero function. In the normalized noncontractive setting, the paper develops a structural transform theory rather than a uniform approximation theorem. A two-point transform is shown to characterize approximate h –convexity, while a finite-support transform gives an explicit representation of the h –convex envelope in the multiplicative case. As applications, approximate Hermite–Hadamard type inequalities, weighted integral estimates, and a composition principle for approximate h –convexity are derived. The results are interpreted for the classical convex, s –convex, and P –function settings.