Let G be a simple connected graph with n vertices and m edges, and sequence of vertex degrees d 1 ≥ d 2 ≥ · · · ≥ d n > 0. If vertices i and j are adjacent, we write i ∼ j. Denote by Π 1 , Π * 1 , Q α and H α the multiplicative Zagreb index, multiplicative sum Zagreb index, general first Zagreb index, and general sum-connectivity index, respectively. These indices are defined as Π 1 = n i=1 d 2 i , Π * 1 = i∼j (d i + d j), Q α = n i=1 d α i and H α = i∼ j (d i + d j) α. We establish upper and lower bounds for the differences H α − m Π * 1 α m and Q α − n (Π 1) α 2n. In this way we generalize a number of results that were earlier reported in the literature.