Submultiplicative functions and operator inequalities

Let T : C¹(ℝ) → C(ℝ) be an operator satisfying the "chain rule inequality" T(f ○ g) ≤ (Tf) ○ g · Tg; f; g ∈ C¹(ℝ): Imposing a weak continuity and a non-degeneracy condition on T, we determine the form of all maps T satisfying this inequality together with T(-Id)(0) < 0. They have the form (Formula presented)., with p > 0, H ∈ C(ℝ), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable functions K on R which are continuous at 0 and 1 and satisfy K(-1) < 0 < K(1). Any such map K has the form (formula presented)., with A ≥ 1 and p > 0. Corresponding statements hold in the supermultiplicative case with 0 < A ≤ 1.