We establish new Bernstein-type inequalities for a linear operator \( \mathscr {T}_g \) acting on complex polynomials, generalizing classical results in geometric function theory. Notably, we introduce an operator \( \mathscr {T}_g \) induced by polynomials \( g \) with all zeros in the open unit disk, whose action is governed by the Schur-Szegő composition theorem and derive sharp bounds relating the maximum norms of \( \mathscr {T}_g[p] \) and \( p \) on the unit disk. Our framework unifies and extends fundamental inequalities of Bernstein, Lax, Ankeny-Rivlin, and Aziz-Dawood for polynomials with restricted zeros. The results not only recover known theorems as corollaries but also yield novel estimates for broader operator classes. This work advances the interplay between operator theory and polynomial inequalities, providing tools applicable in approximation theory and related fields.
