Positive linear operators and inequalities involving convex functions

We study inequalities involving convex functions and positive linear operators and interpret them within the framework of convex stochastic order ≤ cx . Our approach is operatorial in nature and provides a unified view of such inequalities involving several classical approximation processes. By considering a broad class of positive linear operators, we obtain new inequalities that describe dominance between the corresponding induced measures. In addition to the direct problem, showing that the convexity of a function f implies inequalities of the type L n f ≥ M n f , we also consider the inverse problem: given f and a suitable family of measures μ ≤ cx ν satisfying ∫ f d μ ≤ ∫ f d ν , does it follow that f must be convex? This inverse problem is particularly challenging. As an illustrative example, we ask whether, for f ∈ C [0, ∞), the condition S t f ≤ V t f for all t > 0, where S t and V t denote the Szász–Mirakjan and classical Baskakov operators respectively, necessarily implies the convexity of f .