The document presents theorems and corollaries that establish upper bounds for the number of zeros an analytic function can have inside the unit disk, based on restrictions on the function's coefficients. Theorem 1 shows that if the coefficients satisfy certain conditions involving inequalities, then the number of zeros does not exceed a given expression involving the coefficients and disk radius. Several corollaries generalize or simplify the results. The theorems are proved using known results about analytic functions and their zeros.