A dimension function is an increasing, continuous function $%
f:R_{+}\rightarrow R_{+}$ such that $f(r)\rightarrow 0$ as $r\rightarrow 0$.

Say that a dimension function $f$ is *essentially sub-linear* if
there exists $B>1$ such that $\underset{x\rightarrow 0}{\lim \sup }\frac{f\left( Bx\right) }{f\left(x\right) }<B.$

(These functions arise in the *f*-dimensional Hausdorff measure of sets considered in Diophantine approximation).

The above condition does not hold for $f\left( x\right) =x$ or $f\left( x\right) =x\log \left( 1/x\right)$. However it is clearly satisfied for the dimension functions $f\left( x\right) =x^{s}$ when $0\leq s<1$.

I am seeking other elementary function examples or just examples.

Thank you for any interest and assistance.