A dimension function is an increasing, continuous function $% f:R_{+}\rightarrow R_{+}$$% f:\mathbb R_{+}\rightarrow \mathbb R_{+}$ such that $f(r)\rightarrow 0$$f(r)\to 0$ as $r\rightarrow 0$$r\to 0$.
Say that a dimension function $f$ is essentially sub-linear if there exists $B>1$ such that $\underset{x\rightarrow 0}{\lim \sup }\frac{f\left( Bx\right) }{f\left(x\right) }<B.$$\limsup_{x\to 0} \frac{f\left( Bx\right) }{f\left(x\right) }<B.$
(These functions arise in the f$f$-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.
