Proposition 15.32.4.D697A55DE07C The functional $\rho _ h$ satisfies $\rho _ h(tp)=|t|\rho _ h(p)$ for scalars $t\ne 0$, $\rho _ h(p)\leq \rho _ h(q)$ whenever $|p|\leq |q|$, and
\[ \rho _ h(p+q)\leq \rho _ h(p)+\rho _ h(q). \]
In particular, if $b_ h(f),b_ h(g)>0$, then
\[ b_ h(f+g)\geq \bigl (b_ h(f)^{-1/n}+b_ h(g)^{-1/n}\bigr )^{-n}>0. \]
Proof.
By (15.29.2), $\rho _ h(p)=\sup _\gamma L_{h,\gamma }(|p|)$ is a supremum of positive linear functionals evaluated at $|p|$. This gives homogeneity and monotonicity, and, together with $|p+q|\leq |p|+|q|$, the triangle inequality. Finally, $(f+g)^{1/n}\leq f^{1/n}+g^{1/n}$ gives
\[ b_ h(f+g)^{-1/n}=\rho _ h\bigl ((f+g)^{1/n}\bigr )\leq \rho _ h(f^{1/n})+\rho _ h(g^{1/n}) =b_ h(f)^{-1/n}+b_ h(g)^{-1/n}, \]
which is equivalent to the asserted lower bound for $b_ h(f+g)$.
$\square$
Pome
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