Estimates of the number of zeros of some functions with algebraic Taylor coefficients

We prove two theorems about the number of zeros of analytic functions from certain classes that include the Siegel $E$-and $G$-functions. By using these theorems, we arrive at a new proof of the Gel'fond-Schneider theorem and improve the result that the numerical determinant does not vanish in the proof of the Shidlovskii theorem.