Submultiplicative function vs regular variation function

Question

Let us define the following two function class: A submultiplicative function is defined by $f(xy)\leq f(x)f(y), x,y\geq 0$ refer paper 1989(Gustavasson). A regular variation function is defined by $g(\lambda x)/g(x)\to \lambda^\alpha, x,\lambda\geq 0$ refer book 2013(Bingham). As we can see that, if $f(xy)=f(x)f(y)$, we can get $f(x)=x^\beta$ and this kind of fuinction is regular variation. Can we claim that: A submultiplicative function is regular variation function? Is there someone can give me some trick or some reference to check it?

Answer 1

The function $f(t)=t^p(1+|\sin(\log(t)|)$, $p\ge1$ is submultiplicative, but it is not of regular variation (see e.g., Example 5 in Maligranda L. Indices and interpolation. Dissertationes Mathematicae. 1985; 234:1–54).