  We improve the Gagliardo-Nirenberg inequality \[
  \|\varphi\|_{L^q(\mathbb{R}^n)} \le C \|\nabla\varphi\|_{L^r(\mathbb{R}^n)}
\mathcal{L}^{-(\frac 1q - \frac{n-r}{rn})}
(\|\nabla\varphi\|_{L^r(\mathbb{R}^n)}), \] $r=2$, $0<q<\frac{rn}{(n-r)_+}$,
$\mathcal{L}$ generalizing $\mathcal{L}(s)=\ln^{-1}\frac 2s$ for $0<s<1$, from
[M. Fila and M. Winkler: A Gagliardo-Nirenberg-type inequality and its
applications to decay estimates for solutions of a degenerate parabolic
equation, Adv. Math., 357 (2019), https://doi.org/10.1016/j.aim.2019.106823]
for rapidly decaying functions ($\varphi\in
W^{1,r}(\mathbb{R}^n)\setminus\{0\}$ with finite $K=\int_{\mathbb{R}^n}
\mathcal{L}(\varphi)$) by specifying the dependence of $C$ on $K$ and by
allowing arbitrary $r\ge1$.
