Mihai N. Pascu

Scaling coupling of reflecting Brownian motions and the hot spots problem

We introduce a new type of coupling of reflecting Brownian motions in smooth planar domains, called scaling coupling. We apply this to obtain monotonicity properties of antisymmetric second Neumann eigenfunctions of convex planar domains with one line of symmetry. In particular, this gives the proof of the hot spots conjecture for some known types of domains and some new ones.

A probabilistic proof of the fundamental theorem of algebra

We use Levys theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.

Neighbourhoods of univalent functions

The main result shows that a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighbourhood consisting entirely of univalent functions. For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro–Warschawski–Wolff univalence criterion. We also present an application of the main result in terms of Taylor series, and we show that the hypothesis of our main result is sharp. 2000 Mathematics subject classification: primary 30C55; secondary 30C45, 30E10.

Brownian motion with killing and reflection and the "hot-spots" problem

Abstract.We investigate the ‘‘hot–spots’’ property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the ‘‘hot–spots’’ property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other.

Starlike approximations of analytic functions

Abstract When an analytic function is not univalent, it is often of interest to approximate it by univalent functions. In this paper we introduce a measure of the non-univalency of a function and we derive a method for constructing the best starlike univalent approximations of analytic functions with respect to it, suitable for both practical problems and numerical implementation.

Convex approximations of analytic functions

Abstract We introduce a method for constructing the best approximation of an analytic function in a subclass K ∗ ⊂ K of convex functions, in the sense of the L 2 norm. The construction is based on solving a certain semi-infinite quadratic programming problem, which may be of independent interest.

Connections Between the Dirichlet and the Neumann Problem for Continuous and Integrable Boundary Data

We present results concerning the representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. We show that the representation holds in the case of integrable boundary data, thus providing an explicit solution of the generalized solution of the Neumann problem.

A maximum modulus principle for non-analytic functions defined in the unit disk

Maximum principles for analytic functions are very important tools in the analytic function theory. In the present paper, we consider a maximum principle for a class of non-analytic function defined on the open unit disk. Some examples for our theorems are also shown.