I have a question concerning the SERF-reducibility of Impagliazzo, Paturi and Zane and subexponential algorithms. The definition of SERF-reducibility gives the following:
If $P_1$ is SERF-reducible to $P_2$ and there $O(2^{\varepsilon n})$ algorithm for $P_2$ for each $\varepsilon > 0$, then there is $O(2^{\varepsilon n})$ algorithm for $P_1$ for each $\varepsilon > 0$. (Hardness parameter for both problems is denoted by $n$.)
Some sources seem to imply that the following also holds:
If $P_1$ is SERF-reducible to $P_2$ and there $O(2^{o(n)})$ algorithm for $A_2$, then there is $O(2^{o(n)})$ algorithm for $P_1$.
My question is, does this latter claim actually hold and if it does, is there a write-up of the proof somewhere?
As a background, I've been trying to understand the area around the Exponential Time Hypothesis. IPZ define subexponential problems as ones that have $O(2^{\varepsilon n})$ algorithm for each $\varepsilon > 0$, but this apparently is not sufficient in the light of the current knowledge to imply the existence of a subexponential algorithm for the problem. The same gap seems to be present in the SERF reducibility, but I am partially expecting that I am missing something here...