The document summarizes two theorems regarding the zeros of analytic functions whose coefficients are restricted to certain conditions. Theorem 1 generalizes an earlier result by showing that if the coefficients satisfy certain inequalities, the function does not vanish in a specified region, which reduces to a previous result when certain parameters are set to specific values. Theorem 2 further generalizes the results by considering another set of inequalities on the coefficients and proving the function does not vanish in a different region, which also reduces to an earlier theorem under certain conditions. The proofs of the theorems use a lemma and apply Schwarz's lemma to derive the zero-free regions.