# Mini-Workshop Complex Analysis and applications, Faculty of Mathematics, February 27-28, 2012

Mini-Workshop "Complex Analysis and applications", will be held on Monday and Tuesday, 27-28th of February at Faculty of Mathematics, room BIM (4th floor, Computing lab).

The working language of the Workshop is English.

The schedule is as follows:

MONDAY, 27.2, 12h, room BIM

1. Matti Vuorinen, University of Turku, Finland

"Quasihyperbolic geometry and quasiconformal maps"

Abstract: This talk is a survey of my recent work with my coauthors during the past year. The talk is divided into four parts, each part based on a preprint. X. Zhang and G. Wang are PhD students at the University of Turku.

M. Vuorinen and G. Wang: Bisection of geodesic segments in hyperbolic geometry.
arXiv:1108.2948v2 [math. MG]

2. Miodrag Mateljević, Vladimir Božin, Miljan Knežević, Faculty of Mathematics, University of Belgrade

"Quasiconformality of harmonic mappings between smooth Jordan domains"

Abstract: Suppose that $h$ is a harmonic mapping of the unit disc onto a $\displaystyle C^{1,\,\alpha}$ domain $D$. Then $h$ is q.c. if and only if it is bi-Lipschitz. In particular, we consider sufficient and necessary conditions in terms of boundary function that $h$ is q.c. We give a review of recent related results including the case if domain is the upper half plane. We also consider harmonic mapping with respect to $\rho$ metric on codomain.

3. Discussion

TUESDAY, 28.2, 11h, room BIM

1. Stamatis Pouliasis, Aristotle University, Thessaloniki, Greece

"Versions of Schwarz's lemma for condenser capacity and inner radius"

Abstract: We will consider variants of the classical Schwarz's lemma involving monotonicity properties of condenser capacity and inner radius. Also, we will examine when a similar monotonicity property holds for the hyperbolic metric. This is joint work with D. Betsakos.

2. Iason Efraimidis, Aristotle University, Thessaloniki, Greece

"Variations of Schwarz lemma"

Abstract: Suppose that f maps the unit disc D holomorphically into D and f(0)=0. A classical inequality due to Littlewood generalizes Schwarz’s lemma and asserts that every w in f(D) has modulus less or equal to the product of the modulus of its pre-images. We present a similar inequality proved by D.Betsakos, in which the assumption of f(D) being a subset of D is replaced by the weaker assumption Diamf(D)=2. The main tools in the proof are Green’s function and Steiner symmetrization.

3. Vesna Manojlović, Faculty of Organization Sciences, University of Belgrade

"Boundary modulus of continuity and quasiconformal mappings"

Abstract :Let $D$ be a bounded domain in $\mathbb R^n$, $n \geq 2$, and
let $f$ be a continuous mapping of $\overline D$ into $\mathbb R^n$ which
is quasiconformal in $D$. Suppose that $|f(x) - f(y)| \leq \omega(|x-y|)$
for all $x$ and $y$ in $\partial D$, where $\omega$ is a non-negative
non-decreasing function satisfying $\omega(2t) \leq 2\omega(t)$ for
$t \geq 0$. We prove, with an additional growth condition on $\omega$, that
$|f(x) - f(y)| \leq C \max \{\omega(|x-y|), |x-y|^\alpha \}$ for all $x, y \in D$, where $\alpha =K_I(f)^{1/(1-n)}$. The talk is based on [AMN].

[AMN] Miloš Arsenović, Vesna Manojlović and Raimo Nakki,
Boundary modulus of continuity and quasiconformal mappings,
to appear in Ann. Acad. Sci. Fenn.

4. Miloš Arsenović, Faculty of Mathematics, University of Belgrade
(title and abstract will be announced)

5. Discussion