On a measure of irrationality for values of $G$-functions

It is shown that values of $G$-functions satisfying a system of linear differential equations are irrational at rational points $a/b$ with $a\in\mathbb Z$ and $b\in\mathbb N$ such that $b>C(\varepsilon)|a|^{2+\varepsilon}$ for an arbitrary positive $\varepsilon$. In the case of a generalized polylogarithmic function
$$f(z)=\sum_{\nu=1}^\infty\frac{z^\nu}{(\nu+\lambda)^m}, \quad m\geqslant 2, \enskip \lambda\in\mathbb Q\setminus\{-1,-2,\dots\},$$
an explicit form of $C(\varepsilon)$ is found.