Generalized Rolle theorem in ℝ n and ℂ

Abstract

The Rolle theorem for functions of one real variable asserts that the number of zeros off on a real connected interval can be at most that off′ plus 1. The following inequality is a multidimensional generalization of the Rolle theorem: if l[0,1] → ℝ n ,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 thederived curve l′ [0,1], → ℝ n $$t \mapsto \dot x(t)$$ , the following inequality holds: L(l) ⩽ L(l′). For the analytic functionF(z) defined in a neighborhood of a closed plane curve Г ⊂ ℂ ≃ ℝ2 this inequality implies that $$\tilde V$$ Γ(F) ⩽ $$\tilde V$$ Γ(F′) + ϰ(Γ), where $$\tilde V$$ Γ(F) is the total variation of the argument ofF along Γ, and ϰ(Γ) is the integral absolute curvature of Γ. 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 $$\tilde V$$ Γ(F) and another quantity, called theBernstein index, which is expressed in terms of the modulus growth of an analytic function.