The first class of functions is called sublinear, see for instance Proposition 1.1.4 (Convex functions; Hiriart-Urruty, Claude Lemaréchal).
The second class of functions that you mention is essentially the same as the Wright convex functions, see for instance pg. 223 of A.W. Roberts, D.E. Varberg, "Convex functions", Academic Press, 1973.
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