Lower bounds of linear forms of values of $G$ functions

Lower bounds are obtained for linear forms of values of Siegel's $G$ functions. In particular, it is found that if $\alpha_1,\dots,\alpha_m$ are pairwise distinct nonzero rational numbers, then for any positive $\varepsilon$ and a natural $q>q_0(\varepsilon,\alpha_1,\dots,\alpha_m)$ we have for any nonzero set $(x_0,x_1,\dots,x_m)$ of integers the inequality
$$ |x_0+x_1\ln(1+\alpha_1q^{-1})+\dots+x_m\ln(1+\alpha_mq^{-1})|>q^{-\lambda}(h_1\dots h_m)^{-1-\varepsilon}, $$
where $h_i=\max(1,|x_i|)$, and $\lambda=\lambda(\varepsilon,\alpha_1,\dots,\alpha_m)$.