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

We present a version of Rolle's theorem for real exponential polynomials ha ving a number L sufficiently large of zeros in a compact set K of the compl ex plane. We show that the derivative of the exponential polynomials have a t 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 r eal exponential polynomials. Besides, we display a Rolle's theorem for high er-order derivatives and as a conclusion make a few comments about the maxi mal number of zeros a real exponential polynomials may have in a given comp act set of C.. (C) 2001 Editions scientifiques et medicales Elsevier SAS.