Mark Elin

Fractional Iteration and Functional Equations for Functions Analytic in the Unit Disk

We establish criteria for the embeddability of an analytic function into a semigroup of analytic self-mappings of the open unit disk. We do this by studying properties of solutions to the Abel and Schröder functional equations.

Spiral-like functions with respect to a boundary point

Although starlike functions normalized by the condition h(0)=0 have been studied intensively during the last century, until 1981 a few works only dealt with starlike functions with respect to a boundary point.

The Schwarz Lemma: Rigidity and Dynamics

The Schwarz Lemma has given impetus to developments in several areas of complex analysis and mathematics in general. We survey some investigations related to its three parts (invariance, rigidity, and distortion) that began early in the twentieth century and are still being carried out. We consider only functions analytic in the unit disk. Special attention is devoted to the Boundary Schwarz Lemma and to applications of the Schwarz–Pick Lemma and the Boundary Schwarz Lemma to modern rigidity theory and complex dynamics.

Angle distortion theorems for starlike and spirallike functions with respect to a boundary point

We exhibit angle bounds for starlike and spirallike functions with respect to a boundary point. As an application, we obtain a covering theorem for functions convex in one direction.

A rigidity theorem for holomorphic generators on the Hilbert ball

AbstractWe present a rigidity property of holomorphic generators on theopen unit ball B of a Hilbert space H. Namely, if f ∈ Hol(B,H)is the generator of a one-parameter continuous semigroup {F t } t≥0 onB such that for some boundary point τ ∈ ∂B, the admissible limitK-lim z→τf(x)kx−τk 3 = 0, then f vanishes identically on B. Let H be a complex Hilbert space with inner product h·,·i and inducednorm k·k. If H is finite dimensional, we will identify H with C n . We denoteby Hol(D,E) the set of all holomorphic mappings on a domain D ⊂ H whichmap D into a subset E of H, and put Hol(D) := Hol(D,D).We are concerned with the problem of finding conditions for a mappingF ∈ Hol(D,E) to coincide identically with a given holomorphic mappingon D, when they behave similarly in a neighborhood of a boundary pointτ ∈ ∂D.Forholomorphicself-mappings oftheopenunitdisk ∆ := {z ∈ C : |z| < 1},thefollowingresult inthisdirection isduetoD.M.Burns andS.G.Krantz[4].Proposition 1. Let F ∈ Hol(∆) . If the unrestricted limit

Asymptotic Behavior of Semigroups of Holomorphic Mappings

We present several new results on the asymptotic behavior of nonlinear semigroups of holomorphic mappings on the open unit balls of complex Banach and Hilbert spaces.

Filtration of Semi-complete Vector Fields Revisited

In this note we use a technique based on subordination theory in order to continue the study of filtration theory of semi-complete vector fields.

Fixed Points of Pseudo-Contractive Holomorphic Mappings

We study conditions that ensure the existence of fixed points of pseudo-contractive mappings originally considered by Browder, Kato, Kirk and Morales. Specifically we consider holomorphic pseudo-contractions on the open unit ball of a complex Banach space which in general are not necessarily bounded. As a consequence, we obtain sufficient conditions for the existence and uniqueness of the common fixed point of a semigroup of holomorphic self-mappings and study its rate of convergence to this point.

The Radii Problems for Holomorphic Mappings in J*-algebras

The so-called J *-algebras, introduced by L.A. Harris, are closed subspaces of the space L(H) of bounded linear operators over a Hilbert space H which preserves a kind of Jordan triple product structure. The open unit ball of any J *-algebra is a natural generalization of the open unit disk in the complex plane. In particular, any C *-algebra can be realized as a J *-algebra.

Rigidity of Holomorphic Mappings and Commuting Semigroups

The problem of finding conditions for a holomorphic mapping F to coincide identically with a given holomorphic mapping G when they behave similarly on some subset of their common domain of definition, has been studied by many mathematicians.