This paper intends to show some operator and norm inequalities involving synchronous and asynchronous functions. Among other inequalities, it is shown that if A, B ∈ B (H) are two positive operators and f, : J → R are asynchronous functions, then f (A) (A) + f (B) (B) ≤ 1 2 f 2 (A) + 2 (A) + f 2 (B) + 2 (B).