Length functions of lemniscates

Abstract

Abstract. We study metric and analytic properties of generalized lemniscates Et(f ) = {z ∈ C : ln |f (z)| = t}, where f is an analytic function. Our main result states that the length function |Et(f )| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln |Et(f )| is convex on any interval free of critical points of ln |f |. As another application we deduce explicit formulae of the length function in some special cases.

Some Recent Results on the Geometry of Complex Polynomials: The Gauss--Lucas Theorem, Polynomial Lemniscates, Shape Analysis, and Conformal Equivalence

“…conjectured that p n (z) = z n + 1 is the polynomial which maximizes this length: Λ 1 (p n ) = L n . Note note that the maximal length L n is known to be achieved by some polynomial, and that Λ 1 (p n ) is known to equal 2n+O(1) (for these and other results on the so-called Erdős-Herzog-Piranian Lemniscate Problem, see references in [29,14]). In 2006, C. Wang and L. Peng [29] studied the level sets of the proposed maximizing polynomial z n + 1.…”

Polynomials and lemniscates of indefiniteness

“…Of course, then the complement of f −1 (T) has infinitely many components. Occasionally f −1 (T) for an analytic function f is called a generalized lemniscate; see [22]. If f is analytic but not necessarily entire, then (3.9) yields a bound on the speed of convergence in case ⊂ f −1 (T) such that is located in a disk where the Maclaurin series of f converges.…”

The arc length and topology of a random lemniscate

“…The maximum was conjectured [5] to occur for the so-called Erdös lemniscate, i.e, when p(z) = z n − 1. This conjecture remains open but has seen positive results [2,7,12], and Fryntov and Nazarov [8] have proved that Erdös lemniscate is indeed a local maximum and that as n → ∞ the maximum length is 2n + o(n) which is asymptotic to the conjectured extremal.…”