Uniform openness of multiplication in Banach spaces $L_p$

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 ${\mathrm{\ell }}_1\times c_0$ to ${\mathrm{\ell }}_1$. This strengthens the former results obtained by M. Balcerzak, A. Majchrzycki and A. Wachowicz.