An estimate of an incomplete linear form in several algebraic numbers

Let $\mu>m-1$, let $\nu$ be a rational number, and let $\omega_k=b_k^\nu$, where $b_k\ne0$ are distinct numbers of an imaginary quadratic field $K$, which satisfy some additional conditions. Then
\begin{gather*} |x_1\omega_1+\dots+x_m\omega_m|>X^{-\mu},\\ X=\max_{1\leqslant k\leqslant m}|x_k|\geqslant X_0>0,\\ \end{gather*}
where $x_1,\dots,x_m$ are integers of the field $K$, and $X_0$ is an effective constant.