Page 148 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 148
CURRENT TOPICS IN COMPLEX ANALYSIS (MS-32)
Convex sums of biholomorphic mappings and Extension operators in Cn
Eduard Stefan Grigoriciuc, eduard.grigoriciuc@ubbcluj.ro
Babes-Bolyai University of Cluj-Napoca, Romania
Let Bn be the Euclidean unit ball in Cn and let U be the unit disc in C. The aim of this work is
to study convex combinations of biholomorphic mappings on Bn starting from a result proved
by Chichra and Singh in the case of one complex variable. They obtained the conditions in
which a convex combination of the form (1 − λ)f + λg is starlike on U , when f and g are
starlike on the unit disc U and λ ∈ [0, 1]. Using this ideea, we can construct a similar result
for the case of several complex variables. Then, we use this result to characterize convex sums
of biholomorphic starlike mappings on the Euclidean unit ball Bn. Moreover, we obtain some
remarks on convex sums of extension operators defined for locally univalent functions (for
example, the Graham-Kohr extension operator).
Geometry of planar domains and their applications in study of conformal
and harmonic mappings
Stanisława Kanas, skanas@ur.edu.pl
University of Rzeszow, Poland
The famous Riemann Mapping Theorem states that for every simply connected domain Ω = C
containing a point w0 there exists a essentially unique univalent function f such that f (0) = w0
and fz(0) > 0, that maps the unit disk D onto Ω.
Open sense-preserving quasiconformal mappings of D arised as a solutions of linear elliptic
partial differential equations of the form
f z¯(z) = ω(z)fz(z), z ∈ D,
where ω is an analytic function from D into itself, known as a dilatation of f and such that
|ω(z)| < k < 1.
A natural generalization of the classical class of normalized univalent functions on D is the
class of sense-preserving univalent harmonic mappings on D of the form f = h + g normalized
by h(0) = g(0) = h (0) − 1 = 0.
In the context of univalent, quasiconformal and planar harmonic mappings a problem of
convexity, linear convexity, starlikeness, etc. have been intensively studied in the past decades.
Additional properties of a planar domains exhibits a very rich geometric and analytic properties.
We discuss behavior of the function f for which some functional are limited to the Both
leminiscates, Pascal snail, hyperbola and conchoid of the Sluze. Some appropriate examples
are demonstrated.
146
Convex sums of biholomorphic mappings and Extension operators in Cn
Eduard Stefan Grigoriciuc, eduard.grigoriciuc@ubbcluj.ro
Babes-Bolyai University of Cluj-Napoca, Romania
Let Bn be the Euclidean unit ball in Cn and let U be the unit disc in C. The aim of this work is
to study convex combinations of biholomorphic mappings on Bn starting from a result proved
by Chichra and Singh in the case of one complex variable. They obtained the conditions in
which a convex combination of the form (1 − λ)f + λg is starlike on U , when f and g are
starlike on the unit disc U and λ ∈ [0, 1]. Using this ideea, we can construct a similar result
for the case of several complex variables. Then, we use this result to characterize convex sums
of biholomorphic starlike mappings on the Euclidean unit ball Bn. Moreover, we obtain some
remarks on convex sums of extension operators defined for locally univalent functions (for
example, the Graham-Kohr extension operator).
Geometry of planar domains and their applications in study of conformal
and harmonic mappings
Stanisława Kanas, skanas@ur.edu.pl
University of Rzeszow, Poland
The famous Riemann Mapping Theorem states that for every simply connected domain Ω = C
containing a point w0 there exists a essentially unique univalent function f such that f (0) = w0
and fz(0) > 0, that maps the unit disk D onto Ω.
Open sense-preserving quasiconformal mappings of D arised as a solutions of linear elliptic
partial differential equations of the form
f z¯(z) = ω(z)fz(z), z ∈ D,
where ω is an analytic function from D into itself, known as a dilatation of f and such that
|ω(z)| < k < 1.
A natural generalization of the classical class of normalized univalent functions on D is the
class of sense-preserving univalent harmonic mappings on D of the form f = h + g normalized
by h(0) = g(0) = h (0) − 1 = 0.
In the context of univalent, quasiconformal and planar harmonic mappings a problem of
convexity, linear convexity, starlikeness, etc. have been intensively studied in the past decades.
Additional properties of a planar domains exhibits a very rich geometric and analytic properties.
We discuss behavior of the function f for which some functional are limited to the Both
leminiscates, Pascal snail, hyperbola and conchoid of the Sluze. Some appropriate examples
are demonstrated.
146
