Budan’s theorem for a class of entire functions

Abstract

It is easy to see that the even qualification can be omitted if all the zeros of f(z) are real, so that in this case we get exact information on every interval. In this note we describe a class of entire functions for which the natural generalization of Budan's rule likewise gives exact information about the number of zeros in a real interval. This class K is the set of entire functions f(z) which are real for real z, have negative real zeros and positive Taylor coefficients, and are of finite exponential type. Such functions can be written in any of the forms