The main goal of this article is to present new inequalities for (p, h)-convex and (p, h) log-convex functions for a non-negative super-multiplicative and super-additive function h. Our first main result will be hλ ( v µ ) ≤ (h(1 − v) f (a) + h(v) f (b))λ − f λ [ ((1 − v)ap + vbp) 1p ] (h(1 − µ) f (a) + h(µ) f (b))λ − f λ [ ((1 − µ)ap + µbp) 1p ] ≤ hλ ( 1 − v 1 − µ ) , for the positive (p, h)-convex function f ,when λ ≥ 1, p ∈ R\{0} and 0 ≤ v ≤ µ ≤ 1. This gives a generalization of an important result due to M. Sababheh [Linear Algebra Appl. 506 (2016), 588–602]. As applications of our results, we present many inequalities for the trace, and the symmetric norms for τ-measurable operators.