Eremenko’s Conjecture for Functions with Real Zeros: The Role of the Minimum Modulus

We consider the class of real transcendental entire functions $f$ of finite order with only real zeros and show that if the iterated minimum modulus tends to $\infty $, then the escaping set $I(\,f)$ of $f$ has the structure of a spider’s web, in which case Eremenko’s conjecture holds. This minimum modulus condition is much weaker than that used in previous work on Eremenko’s conjecture. For functions in this class, we analyse the possible behaviours of the iterated minimum modulus in relation to the order of the function $f$.

On a Theorem of Schwarz Type for Quasiconformal Mappings in Space

A space ring R is defined as a domain whose complement in the Moebius space consists of two components. The modulus of R can be defined in variously different but essentially equivalent ways (see e.g. Gehring [3] and Krivov [5]), which is denoted by mod R. Following Gehring [2], we refer to a homeomorphism y(x) of a space domain D as a k-quasiconformal mapping, if the modulus conditionis satisfied for all bounded rings R with their closure , where y(R) denotes the image of R by y = y(x). Then, it is evident that the inverse of a k-quasi-conformal mapping is itself k-quasiconformal and that a k1-quasiconformal mapping followed by a k2-quasiconformal one is k1k2-quasiconformal. It is also well known that the restriction of a Moebius transformation to a space domain is equivalent to a 1-quasiconformal mapping of its domain.