Dov Aharonov

A minimal area problem in conformal mapping

LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU such thatf(0)=f′(0)−1=0. The main result of the paper is that area (f(U) ≥27π/7)(2-α)−2 for allf∈S such that |f″(0)|=2α, 1/2<α<2. This solves a long-standing extremal problem for the class of functions considered.

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 Dynamics of a Piecewise Linear Map and its Smooth Approximation

The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fe: (x, y) → (1 - y - fe(x), x), where fe(x) is real analytic and close to |x| for small values of e. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fe, provided that e is small.

Extremal Problems for Nonvanishing Functions in Bergman Spaces

In this paper, we study general extremal problems for non-vanishing functions in Bergman spaces. We show the existence and uniqueness of solutions to a wide class of such problems. In addition, we prove certain regularity results: the extremal functions in the problems considered must be in a Hardy space, and in fact must be bounded. We conjecture what the exact form of the extremal function is. Finally, we discuss the specific problem of minimizing the norm of non-vanishing Bergman functions whose first two Taylor coefficients are given.

Proof of the bieberbach conjecture for a certain class of univalent functions

In the following we prove that for a given univalent function such that |a2| <0.867, |an|≦n for eachn. The method of proof is closely related to Milin’s method.

The sharp constant in the ring lemma

In his paper [2] Lowell J. Hansen found a (nonlinear) recurrence formula for the (sharp constant appearing in the “Ring Lemma” of Rodin and Sullivan [3]. In the following we improve Hansens result and replace his recurrence relation by a linear recurrence formula leading to a closed formula for the Ring Lemma constant. Moreover, we show that the Ring Lemma constant is a reciprocal of an integer for each n

Discrete Sturm comparison theorems on finite and infinite intervals

The Sturm comparison theorem for second-order Sturm–Liouville difference equations on infinite intervals is established and discussed.

On the Bieberbach conjecture for functions with a small second coefficient

In the following we prove that for a given univalent function such that |a2|<1.05, |an|<n for eachn. This is an improvement of the result in [1].

A Binomial Identity via Differential Equations

Abstract In the following we discuss a well-known binomial identity. Many proofs by different methods are known for this identity. Here we present another proof, which uses linear ordinary differential equations of the first order.

Minimal area problems for functions with integral representation

We study the minimization problem for the Dirichlet integral in some standard classes of analytic functions. In particular, we solve the minimal areaa2-problem for convex functions and for typically real functions. The latter gives a new solution to the minimal areaa2-problem for the classS of normalized univalent functions in the unit disc.