On the Existence of Real Entire Functions with a Prescribed Ordered Set of Stationary Values

Abstract

To each ordered set of real stationary values (with multiplicities) is shown to correspond a real entire function ƒ taking on the stationary values in the given order and with the given multiplicities along the real axis. This function has a derivative ƒ′, whose roots are all real, and which is of the Pólya-Laguerre class; the function ƒ is, apart from an arbitrary real affine transformation (conserving order) of the independent variable, the only function satisfying all of the conditions above.