Nicolae R. Pascu

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.

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.