Estimates from below of polynomials in the values of analytic functions of a certain class

Estimates from below are obtained for polynomials with integral coefficients in the values of certain Siegel $G$-functions at the algebraic points of a special form. In particular, it is proved that if $\alpha_1,\dots,\alpha_s$ ($\alpha_1\cdots\alpha_s\ne0$) are pairwise distinct algebraic numbers, $q$ is a natural number, and $P(x_1,\dots,x_s)\not\equiv0$ is a polynomial with integral coefficients of degree not greater than $d$ and height not exceeding $H$, then for $q>q_0(d,\alpha_1,\dots,\alpha_s)$ we have
$$\Bigl|P\Bigl(\ln\Bigl(1+\frac{\alpha_1}q\Bigr),\dots,\ln\Bigl(1+\frac{\alpha_s}q\Bigr)\Bigr)\Bigr|>q^{-\lambda}H^{-\mu}, $$
where the constants $q_0$ and $\mu$ can be effectively computed.
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