A Rolle's theorem for real exponential polynomials in the complex domain

Abstract

We present a version of Rolle's theorem for real exponential polynomials having a number L sufficiently large of zeros in a compact set K of the complex plane. We show that the derivative of the exponential polynomials have at least L−1 zeros in a region slightly larger than K . The method of proof is elementary and similar to that of the classical Jensen's theorem about the location of the zeros of the derivative of a real polynomial. The proof also relies on known results concerning the distribution of the zeros of real exponential polynomials. Besides, we display a Rolle's theorem for higher-order derivatives and as a conclusion make a few comments about the maximal number of zeros a real exponential polynomials may have in a given compact set of C .