In the functional-analytic sense used here, a superlinear functional is real valued, positively homogeneous and superadditive. Equivalently is a sublinear functional. This definition concerns functional inequalities, rather than a comparison of asymptotic growth rates. Such a is concave, satisfies , and can serve as a lower support in the sandwich form of the Hahn-Banach theorem.
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A **sublinear function** is a function that grows slower than a linear function as its input increases. In mathematical terms, a function \( f(x) \) is considered sublinear if it satisfies the condition: \[ \lim_{x \to \infty} \frac{f(x)}{x} = 0 \] This means that as \( x \) becomes very large, the ratio \( \frac{f(x)}{x} \) approaches 0.