Second-Order Differential Inequality Convexity

Some equivalent statements and basic properties for the generalized convexity assumption p(x)f(x)+q(x)f′(x)+f′′(x)≥0 are proved. Then based on these conclusions, various Jensen type inequalities under the generalized convexity are established. The idea is to transform such p(x),q(x)-convex functions to some simpler p(x),0-convex or 0,q(x)-convex functions, or even convex functions. Ky Fan and Wang-Wang type inequalities are also generalized as applications.