Convex-like inequality, homogeneity, subadditivity, and a characterization of ${\displaystyle L^p}$-norm

Let a and b be fixed real numbers such that 0 < min{a,b} < 1 < a + b. We prove that every function f:(0,∞) → ℝ satisfying f(as + bt) ≤ af(s) + bf(t), s,t > 0, and such that ${\displaystyle lim{\supset }_{t\to 0+}f\left(t\right)\le 0}$ must be of the form f(t) = f(1)t, t > 0. This improves an earlier result in [5] where, in particular, f is assumed to be nonnegative. Some generalizations for functions defined on cones in linear spaces are given. We apply these results to give a new characterization of the ${\displaystyle L^p}$-norm.