This paper studies $K$-sublinear inequalities, a class of inequalities with strong relations to K-minimal inequalities for disjunctive conic sets. We establish a stronger result on the sufficiency of $K$-sublinear inequalities. That is, we show that when $K$ is the nonnegative orthant or the second-order cone, $K$-sublinear inequalities together with the original conic constraint are always sufficient for the closed convex hull description of the associated disjunctive conic set. When $K$ is the nonnegative orthant, $K$-sublinear inequalities are tightly connected to functions that generate cuts--so called cut-generating functions. In particular, we introduce the concept of relaxed cut-generating functions and show that each $R^n_+$-sublinear inequality is generated by one of these. We then relate the relaxed cut-generating functions to the usual ones studied in the literature. Recently, under a structural assumption, Cornuejols, Wolsey and Yildiz established the sufficiency of cut-generating functions in terms of generating all nontrivial valid inequalities of disjunctive sets where the underlying cone is nonnegative orthant. We provide an alternate and straightforward proof of this result under the same assumption as a consequence of the sufficiency of $R^n_+$-sublinear inequalities and their connection with relaxed cut-generating functions.