Vishnyakova A. Polynomials with restrictions on the location of roots and the limiting classes of entire functions

We develop a new theory and find new methods for studying connections between the zero location of polynomials and entire functions and properties of the coefficients of these functions. We also find descriptions and investigate properties of linear operators that preserve hyperbolicity, stability and positivity. We obtain a sufficient condition for total (multiple) positivity of real matrices and prove that this condition is sharp in the class of Hankel matrices and in the class of Toeplitz matrices. New sufficient conditions for stability of complex polynomials and entire functions are obtained; they are shown to be sharp. We obtain a description of the extremal rays of the cone of polynomials with nonnegative coefficients that are nonnegative on the real axis and have degrees less than a given number. A description of the class of linear operators that act diagonally in the standard monomial basis and preserve this cone is found. We obtain a characterization of the zero sets of entire absolutely monotonic functions under condition that these sets do not intersect an angle that contains the negative half-axis. We found the parameter values for which the partial theta-function and its Taylor sections belong to the Laguerre-Polya class. We investigate some other important entire functions and their Taylor sections, and determine whether these functions belong to the Laguerre-Polya class. We find a new characterization of the Laguerre-Polya class in terms of generalized Laguerre inequalities. A new sharp estimate from below for the mesh of the image of a hyperbolic polynomial under central finite-difference linear operator with constant coefficients is obtained. We find a description of linear finite-difference operators with constant coefficients that preserve the set of hyperbolic polynomials and that preserve the set of polynomials whose zeros are contained in the horizontal strip. We establish important properties of the zeroes of the image of an entire function from the Laguerre-Polya class under central finite-difference linear operator. We obtain a description of linear finite-difference operators, whose coefficients are entire functions, and which preserve the Laguerre-Polya class of entire functions.