Polynomial-bound join-transitively subnormal subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed polynomial-bound join-transitively subnormal in $$G$$ if there exists a polynomial $$f$$ with integer coefficients such that if $$K$$ is a $$k$$-defining ingredient::subnormal subgroup of $$G$$, $$\langle H, K \rangle$$ (the join of subgroups) is a $$f(k)$$-subnormal subgroup of $$G$$.

Here, a $$k$$-subnormal subgroup is a subgroup whose defining ingredient::subnormal depth is at most $$k$$.

Stronger properties

 * Weaker than::Normal subgroup: The polynomial is $$f(k) = k$$.
 * Weaker than::2-subnormal subgroup: The polynomial is $$f(k) = 2k$$.
 * Weaker than::Linear-bound join-transitively subnormal subgroup: The polynomial is linear in this case.
 * Weaker than::Subnormal-permutable subnormal subgroup:

Weaker properties

 * Stronger than::Join-transitively subnormal subgroup