Proposition 7

If $f$ and $g$ are two functions in $\mathscr{H}(\mathfrak{F}, V)$ then the relation $f - g \ll f$ is equivalent to $f - g \ll g$.

Indeed, $f - g \ll f$ implies that for every $\varepsilon > 0$ there exists an $X \in \mathfrak{F}$ such that $\|f(x) - g(x)\| \leq \varepsilon \|f(x)\|$ for every $x \in X$. But then $(1 - \varepsilon) \|f(x)\| \leq \|g(x)\|$, and consequently $f \preccurlyeq g$, whence (V, p. 215, prop. 4) $f - g \ll g$.