On the zeros of certain composite polynomials and an operator preserving inequalities

If all the zeros of n th degree polynomials f ( z ) and g ( z ) = ∑ k = 0 n λ k n k z k respectively lie in the cricular regions | z | ≤ r and | z | ≤ s | z - σ | , s > 0 , then it was proved by Marden (Geometry of polynomials, Math Surveys, No. 3, American Mathematical Society, Providence, 1949, p. 86) that all the zeros of the polynomial h ( z ) = ∑ k = 0 n λ k f ( k ) ( z ) ( σ z ) k k ! lie in the circle | z | ≤ r max ( 1 , s ) . In this paper, we relax the condition that f ( z ) and g ( z ) are of the same degree and instead assume that f ( z ) and g ( z ) are polynomials of arbitrary degree n and m , respectively, m ≤ n , and obtain a generalization of this result. As an application, we also introduce a linear operator which preserves Bernstein type polynomial inequalities.