Length functions of lemniscates (Q1434247)

The authors study metric and analytic properties of generalized lemniscates \(E_t(f)=\{z\in{\mathbb C}:\ln| f(z)| =t\}\), where \(f\) is an analytic function. Main result states that the length function \(| E_t(f)| \) is a bilateral Laplace transform of a certain positive measure. Let \(({\mathcal U},f, {\mathcal I}\) be a regular lemniscate region. Given an analytic function \(w(z)\) in \(\mathcal U\) there exists a non-decreasing function \(\sigma(x)\) such that \[ L_w(t)=\int\limits_{E_{{\mathcal U},f}(t)}| w(z)| ^2| dz| =\int\limits_{-\infty}^{+\infty}e^{xt}d\sigma(x), \;t\in(\alpha;\beta)\,, \] and the latter integral converges for \(t\in\mathcal I\). In particular, the function \(\ln| E_t(f)| \) is convex on any interval free of critical points of \(\ln| f| \). As another application the authors deduce explicit formulae of the length function in some special cases.