A Complex Analogue of the Rolle Theorem and Polynomial Envelopes of Irreducible Differential Equations in the Complex Domain

We prove a complex analytic analogue of the classical Rolle theorem asserting that the number of zeros of a real smooth function can exceed that of its derivative by at most 1. This result is used then to obtain upper bounds for the number of complex isolated zeros of : (1) functions defined by linear ordinary differential equations (in terms of the magnitude of the coefficients of the equations) ; (2) elements from the polynomial envelope of a linear differential equation with an irreducible monodromy group (in terms of the degree of the envelope) ; (3) successive derivatives of a function defined by a linear irreducible equation (in terms of the order of the derivative). These results generalize the bounds from [2, 5, 6] that were previously obtained for the number of real isolated zeros.