Is there a standard term for the quasi-norm $$\|f\|_{[k]}=\sum_{i=1}^k(\sup\|f^{(i)}\|)^{1/i}$$ ?

It is useful due to the fact that it is reasonably compatible with post-composition by smooth functions: $$\|F(f)\|_{[k]}\leq\mathrm{const}_{F,k}\cdot\|f\|_{[k]}$$ in contrast to the usual $C^k$ norm for which this inequality does not hold.