David Minda

Area, capacity and diameter versions of Schwarz’s Lemma

The now canonical proof of Schwarz’s Lemma appeared in a 1907 paper of Caratheodory, who attributed it to Erhard Schmidt. Since then, Schwarz’s Lemma has acquired considerable fame, with multiple extensions and generalizations. Much less known is that, in the same year 1907, Landau and Toeplitz obtained a similar result where the diameter of the image set takes over the role of the maximum modulus of the function. We give a new proof of this result and extend it to include bounds on the growth of the maximum modulus. We also develop a more general approach in which the size of the image is estimated in several geometric ways via notions of radius, diameter, perimeter, area, capacity, etc.

A multi-point Schwarz-Pick Lemma

We state and prove a general version of the Schwarz-Pick Lemma that involves more than two points in the hyperbolic plane and with appears to contain all known variations of the classical result. We give some applications to complex function theory and then discuss our result in the context of the Pick-Nevanlinna Theorem.

A reflection principle for the hyperbolic metric and applications to geometric function theory

We establish a reflection principle for the hyperbolic metric which has applications to geometric function theory. For instance, the reflection principle yields a number of monotonicity properties of the hyperbolic metric. The sharp form of Landaus Theorem is an immediate consequence of one of these monotonicity properties. The second main application is an interpretation of the reflection principle in terms of convexity relative to hyperbolic geometry.

Quasi-invariant domain constants

Five domain constants are studied in our paper, all related to the hyperbolic geometry in hyperbolic plane regions which are uniformly perfect (in Pommerenke’s terminology). Relations among these domain constants are obtained, from which bounds are derived for the variance ratio of each constant under conformal mappings of the regions, and we also show that each constant may be used to characterize uniformly perfect regions.

Two-point distortion for univalent functions

Abstract We discuss two-point distortion inequalities for (not necessarily normalized) univalent functions f on the unit disk D . By a two-point distortion inequality we mean an upper or lower bound on the Euclidean distance |f(a)−f(b)| in terms of d D (a,b) , the hyperbolic distance between a and b, and the quantities (1−|a| 2 )|f′(a)|, (1−|b| 2 )|f′(b)| . The expression (1−|z|2)|f′(z)| measures the infinitesimal length distortion at z when f is viewed as a function from D with hyperbolic geometry to the complex plane C with Euclidean geometry. We present a brief overview of the known two-point distortion inequalities for univalent functions and obtain a new family of two-point upper bounds that refine the classical growth theorem for normalized univalent functions.

DISTORTION THEOREMS FOR LOCALLY UNIVALENT BLOCH FUNCTIONS

A holomorphic functionf defined on the unit disk d is called a Bloch function provided {fx73-02} For α ∃ (0,1] letB∞(α)denote the class of locally univalent Bloch functionsf normalized by ∥f∥B ≤1f(0) = 0 andf’(0) = α. A type of subordination theorem is established for B∞(α). This subordination theorem is used to derive sharp growth, distortion, curvature and covering theorems for B∞(α).

Triangles, Ellipses, and Cubic Polynomials

(2008). Triangles, Ellipses, and Cubic Polynomials. The American Mathematical Monthly: Vol. 115, No. 8, pp. 679-689.

Euclidean linear invariance and uniform local convexity

Let S(p) be the family of holomorphic functions / defined on the unit disk D , normalized by /(0) = f{0) -1= 0 and univalent in every hyperbolic disk of radius p. Let C(p) be the subfamily consisting of those functions which are convex univalent in every hyperbolic disk of radius p . For p = oo these become the classical families 5 and C of normalized univalent and convex functions, respectively. These families are linearly invariant in the sense of Pommerenke; a natural problem is to calculate the order of these linearly invariant families. More precisely, we give a geometrie proof that C(p) is the universal linearly invariant family of all normalized locally schlicht functions of order at most coth(2/>). This gives a purely geometric interpretation for the order of a linearly invariant family. In a related matter, we characterize those locally schlicht functions which map each hyperbolically &-convex subset of D onto a euclidean convex set. Finally, we give upper and lower bounds on the order of the linearly invariant family S(p) and prove that this class is not equal to the universal linearly invariant family of any order.

The minimum points of the hyperbolic metric

We consider two main problems. гirst, what are the properties of the set E(Ω) of points where the density λω(z) of the hyperbolic metric on a hyperbolic plane domain Ω attains its absolute minimum. Second, what are the properties of the subset D(Ω) (D S (Ω)) of a domain Ω where all holomorphic (meromorphic) self-mappings of Ω are distance-decreasing relative to euclidean (spherical) geometry. For a simply connected proper subdomain of the plane, the two sets E(Ω) and D(Ω) coincide, while on general hyperbolic domains, the second contains the first, and may strictly contain it. The set D(Ω) can be empty, while D s (Ω) is always nonempty. We give conditions both for the existence of minimum points and for the discreteness of the set of minimum points. We show that minimum points of the hyperbolic density cannot get too close to the boundary in a uniform sense.

Möbius invariant metrics bilipschitz equivalent to the hyperbolic metric

We study three Möbius invariant metrics, and three affine invariant analogs, all of which are bilipschitz equivalent to the Poincaré hyperbolic metric. We exhibit numerous illustrative examples.