Page 146 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 146
CURRENT TOPICS IN COMPLEX ANALYSIS (MS-32)
Recent studies on the image domain of starlike functions
Sahsene Altinkaya, sahsenealtinkaya@beykent.edu.tr
Beykent University, Faculty of Arts and Sciences, Department of Mathematics,
34500, Istanbul, Turkey
The geometric properties of image domain of starlike functions is of prominent significance to
present a comprehensive study on starlike functions. For this purpose, by using a relation of
subordination, we investigate a family of starlike functions in the open unit disc
U = {z ∈ C : |z| < 1} .
Subsequently, we discuss some interesting geometric properties, radius problems, general co-
efficients for this class. Further, we point out several recent studies related to image domain of
starlike functions.
Extremal decomposition of the complex plane
Iryna Denega, iradenega@gmail.com
Institute of mathematics of the National Academy of Sciences of Ukraine, Ukraine
The talk is devoted to a few well-known an extremal problems in geometric function theory of a
complex variable associated with estimates of the functionals defined on the systems of mutually
non-overlapping domains [1–9]. An improved method is proposed for solving problems on
extremal decomposition of the complex plane. And effective upper estimates for maximum of
the products of inner radii of mutually non-overlapping domains for any fixed systems of points
of the complex plane at all possible values of some parameter γ are obtained [6–9]. Also we
established conditions under which the structure of points and domains is not important. In
particular, we obtained full solution of an open problem about extremal decomposition of the
complex plane with two free poles located on the unit circle [8].
Problem. [1] Consider the product
n
rγ (B0, 0) r (Bk, ak) ,
k=1
where B0,..., Bn, n 2, are pairwise non-overlapping domains in C, a0 = 0, |ak| = 1, k = 1, n
and γ ∈ (0, n] (r(B, a) be an inner radius of the domain B ⊂ C relative to a point a ∈ B). For
all values of the parameter γ ∈ (0, n] to show that it attains its maximum at a configuration of
domains Bk and points ak possessing rotational n-symmetry.
The proof is due to Dubinin for γ = 1 [1] and to Kuz’mina [2] for 0 < γ < 1. Subsequently,
Kovalev [3] solved this problem under the additi√onal assumption that the angles between neigh-
bouring line segments [0, ak] do not exceed 2π/ γ.
Let γ √γ 2√γ
√nγ
In0(γ) = 4n 4γ n 1 − n .
n2 1 +
n 1 − γ n+ γ
n2 n
Theorem 1. [8] Let γ ∈ (1, 2]. Then, for any different points a1 and a2 of the unit circle
and any mutually non-overlapping domains B0, B1, B2, a1 ∈ B1 ⊂ C, a2 ∈ B2 ⊂ C, a0 = 0 ∈
144
Recent studies on the image domain of starlike functions
Sahsene Altinkaya, sahsenealtinkaya@beykent.edu.tr
Beykent University, Faculty of Arts and Sciences, Department of Mathematics,
34500, Istanbul, Turkey
The geometric properties of image domain of starlike functions is of prominent significance to
present a comprehensive study on starlike functions. For this purpose, by using a relation of
subordination, we investigate a family of starlike functions in the open unit disc
U = {z ∈ C : |z| < 1} .
Subsequently, we discuss some interesting geometric properties, radius problems, general co-
efficients for this class. Further, we point out several recent studies related to image domain of
starlike functions.
Extremal decomposition of the complex plane
Iryna Denega, iradenega@gmail.com
Institute of mathematics of the National Academy of Sciences of Ukraine, Ukraine
The talk is devoted to a few well-known an extremal problems in geometric function theory of a
complex variable associated with estimates of the functionals defined on the systems of mutually
non-overlapping domains [1–9]. An improved method is proposed for solving problems on
extremal decomposition of the complex plane. And effective upper estimates for maximum of
the products of inner radii of mutually non-overlapping domains for any fixed systems of points
of the complex plane at all possible values of some parameter γ are obtained [6–9]. Also we
established conditions under which the structure of points and domains is not important. In
particular, we obtained full solution of an open problem about extremal decomposition of the
complex plane with two free poles located on the unit circle [8].
Problem. [1] Consider the product
n
rγ (B0, 0) r (Bk, ak) ,
k=1
where B0,..., Bn, n 2, are pairwise non-overlapping domains in C, a0 = 0, |ak| = 1, k = 1, n
and γ ∈ (0, n] (r(B, a) be an inner radius of the domain B ⊂ C relative to a point a ∈ B). For
all values of the parameter γ ∈ (0, n] to show that it attains its maximum at a configuration of
domains Bk and points ak possessing rotational n-symmetry.
The proof is due to Dubinin for γ = 1 [1] and to Kuz’mina [2] for 0 < γ < 1. Subsequently,
Kovalev [3] solved this problem under the additi√onal assumption that the angles between neigh-
bouring line segments [0, ak] do not exceed 2π/ γ.
Let γ √γ 2√γ
√nγ
In0(γ) = 4n 4γ n 1 − n .
n2 1 +
n 1 − γ n+ γ
n2 n
Theorem 1. [8] Let γ ∈ (1, 2]. Then, for any different points a1 and a2 of the unit circle
and any mutually non-overlapping domains B0, B1, B2, a1 ∈ B1 ⊂ C, a2 ∈ B2 ⊂ C, a0 = 0 ∈
144
