Geodesic
Lower bounds of linear forms of values of $G$ functions
Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 541-552 
A. I. Galochkin. Lower bounds of linear forms of values of $G$ functions. Matematičeskie zametki, Tome 18 (1975) no. 4, pp. 541-552. http://geodesic.mathdoc.fr/item/MZM_1975_18_4_a6/
@article{MZM_1975_18_4_a6,
     author = {A. I. Galochkin},
     title = {Lower bounds of linear forms of values of $G$ functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {541--552},
     year = {1975},
     volume = {18},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_18_4_a6/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\lambda=\lambda(\varepsilon,\alpha_1,\dots,\alpha_m)$.