generalization of certain well known polynomial inequalities

In this paper, we investigate a generalized class of complex polynomials of the form g(z) = anz n + ∑n v=t an−vz n−v , 1 ≤ t ≤ n, whose zeros are constrained to lie within or on the boundary of the disk |z| ≤ k, k ≤ 1. we establish new bernstein-type inequalities for such polynomials, extending and improving earlier results due to jain [6], aziz and rather [4]. several known inequalities follow as special cases, and the results have implications for the growth estimates and derivative bounds of analytic functions within specified domains.