The Rolle theorem for functions of one real variable
asserts that mthe number of zeros of f on a real connected
interval can be at most that of f' plus 1. The following
inequality is a multidimensional generalization of the Rolle
theorem: if l:[0,1]-> Rn, t-> x(t),
is a closed smooth spatial curve and L(l) is the length
of its spherical projection on a unit sphere, then for the
derived curve l':[0,1]-> Rn, t->
dx(t)/dt, the following inequality holds:
L(l)< L(l'). For the analytic function F(z) defined
in a neighborhood of a closed plane curve G in the complex
plane C=R2
this inequality implies that VG(F)<
VG(F')+K(G), where
VG(F) is the total variation of the argument
of F along G, and K(G) is the integral
absolute curvature of G.
As an application of this inequality, we find an upper bound
for the number of complex isolated zeros of quasipolynomials.
We also establish a two-sided inequality between the variation
index VG(F) and another quantity, called
Bernstein index, which is expressed in terms of the
modulus growth of an analytic function.