Abstract. Let p be a positive number and h a function on R + satisfying h(xy) ≥ h(x)h(y) for any x, y ∈ R + . A non-negative continuous function f onholds for all positive semidefinite matrices A, B of order n with spectra in K, and for any α ∈ (0, 1).In this paper, we study properties of operator (p, h)-convex functions and prove the Jensen, Hansen-Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator (p, h)-convex. In applications, we obtain Choi-Davis-Jensen type inequality for operator (p, h)-convex functions and a relation between operator (p, h)-convex functions with operator monotone functions.