Non-linear coefficient-wise stability and hyperbolicity preserving transformations

Abstract

We study the operation of replacing the coeffcients of a real function with a non-linear combination of its coeffcients. We are particularly interested in the coeffcient-wise transformations that preserve the location of zeros in a prescribed region. The earliest example of such transformations may be traced back to the classical composition theorem of Malo, Schur, and Szego, from which it follows that squaring all the coeffcients of a real entire function with only real zeros preserves the reality of zeros. A next and quite recent example of such transformations is due to P. Branden who studied the operation that subtracts from the square of a coeffcient the product of its two closest neighbors. We relate this form of coeffcient-wise transformation to a system of inequalities characterizing the Laguerre-Polya class of real entire functions. Then using Branden's criterion for stability preserving coeffcient-wise transformations we describe a class of non-linear transformations that arise naturally from this system of inequalities. Other systems of inequalities characterizing the Laguerre-Polya class would potentially yield coeffcient-wise transformations of a different character. We investigate the stability preserving properties of coeffcient-wise transformations that arise from the systems of inequalities characterizing the Laguerre-Polya class. Applications to logarithmic concavity and decreasing sequences yield novel results on the stability preserving properties of these transformations. Under certain conditions the stability-preserving properties of the coeffcient-wise transformations under consideration extend to sequences of real stable polynomials. We highlight several open problems and propose conjectures regarding the stability-preserving properties of transformations considered in this work.