A Rolle's theorem for real exponential polynomials in the complex domain
Let \(g(z)=\sum_{j=1}^np_j(z)\exp(\alpha_j z)\), where \(0\leq \alpha_1<\cdots <\alpha_n\), and \(p_j(z)\) is a real polynomial of degree \(m_j\). Define the degree of \(g\) by \(\deg g:=n-1+\sum_{j=1}^{n-1}m_j\). In this paper, the author derives an analog of Rolle's theorem for exponential polynomials, \(g\), having \(L\geq \deg g\) zeros in a compact region, \(K\), containing the origin. When the degree of \(g\) is sufficiently large, the author shows that at least \(L-1\) zeros of \(g'(z)\) lie in a region slightly larger than \(K\). Rolle's theorem for higher order derivatives is presented in Section 3.
scientific article; zbMATH DE number 1770831