  We show that multiplication from $L_p\times L_q$ to $L_1$ (for $p,q\in
[1,\infty]$, $1/p+1/q=1$) is a uniformly open mapping. We also prove the
uniform openness of the multiplication from $\ell_1\times c_0$ to $\ell_1$.
This strengthens the former results obtained by M. Balcerzak, A. Majchrzycki
and A. Wachowicz.
