Application of Gauss-Lucas theorem to the inequalities for polynomials

This work is a part of recent wave of studies on inequalities which relate the uniform-norm of a complex polynomial to its derivative on the unit disk in the plane. When there is a limit on the zeros of a polynomial, we develop some generalized inequalities that relate the uniform-norm of the polynomial to its derivative. In this paper, we use the zero-preserving property of the derivative on the space of complex polynomials and thereby estimate some Bernstein-type inequalities for polynomials.