It is known (see [4, Brändén, Lemma 2.7]) that a necessary condition for T := Q k (x)D k to be hyperbolicity preserving is that Q k (x) and Q k−1 (x) have interlacing zeros. We characterize all quadratic linear operators, as a consequence we find several classes of Pn-multiplier sequence.Our investigation involves such operators that act on polynomials, in particular, we are interested in polynomials with the following property.whose zeros are all real is said to be hyperbolic. Following the convention of G. Pólya and J. Schur [10, p.89], the constant 0 is also deemed to be hyperbolic. Definition 2. A linear operator T : R[x] → R[x] is said to preserve hyperbolicity (or T is a hyperbolicity preserver ) if T [f (x)] is a hyperbolic polynomial, whenever f (x) is a hyperbolic polynomial.Hyperbolicity preserving operators have been studied by virtually every author who has studied hyperbolic polynomials (see [5] and the references contained therein). The focus of our investigation involves the relationship between hyperbolicity preserving operators and hyperbolic polynomials with interlacing zeros.

P. Bra¨nde´n recently proved a conjecture due to S. Fisk, R.P. Stanley, P.R.W. McNamara and B.E. Sagan. In addition, P. Bra¨nde´n gave a partial answer to a question posed by S. Fisk regarding the distribution of zeros of polynomials under the action of certain non-linear operators. In this article, we give an extension to a result of P. Bra¨nde´n, and also answer a question posed by S. Fisk.

In addition, P. Brändén gave a partial answer to a question posed by S. Fisk regarding the distribution of zeros of polynomials under the action of certain non-linear operators. In this paper, we give an extension to a result of P. Brändén, and we also answer a question posed by S. Fisk.