Relations among starlikeness, convexity and k-fold symmetry of locally biholomorphic mappings in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">C</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:math>

Complex Anal. Oper. Theory

2022

Self Cite

It is known that the starlikeness plays a central role in complex analysis, similarly as the convexity in functional analysis. However, if we consider the biholomorphisms between domains in $${\mathbb {C}}^{n},$$ C n , apart from starlikeness of domains, various symmetries are also important. This follows from the Poincaré theorem showing that the Euclidean unit ball is not biholomorphically equivalent to a polydisc in $${\mathbb {C}}^{n},n>1$$ C n , n > 1 . From this reason the second author in 2003 considered some families of locally biholomorphic mappings defined in the Euclidean open unit ball using starlikeness factorization and a notion of k-fold symmetry. The 2017 paper of both authors contains some results on the absorption by a family $$S(k),k\ge 2,$$ S ( k ) , k ≥ 2 , of the above kind, the families of mappings biholomorphic starlike (convex) and vice versa. In the present paper there is given a new sufficient criterion for a locally biholomorphic mapping f, from the Euclidean ball $${\mathbb {B}}^{n}$$ B n into $${\mathbb {C}}^{n},$$ C n , to belong to the family $$S(k),k\ge 2.$$ S ( k ) , k ≥ 2 . The result is obtained using an n-dimensional version of Jack’s Lemma.

Section: Moreover If Dqmentioning

confidence: 99%

Section: Moreover If Dqmentioning

confidence: 99%

“…Another interesting property of the families S(k) is the fact [3] that the families S(k) and the family Sc of biholomorphic mappings f :…”

Section: The Equality S(2) = St S Is a Good Geometrical Interpretatio...mentioning

confidence: 99%

Section: The Equality S(2) = St S Is a Good Geometrical Interpretatio...mentioning

confidence: 99%

See 2 more Smart Citations

On an Application of Jack’s Lemma, Starlikeness and k-Fold Symmetry in $${\mathbb {C}}^{n}$$

Complex Anal. Oper. Theory

2022

Self Cite

Some results of Fekete–Szegö type for Bavrin’s families of holomorphic functions in $${\mathbb {C}}^{n}$$

2021

Annali di Matematica

Self Cite

In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which elements satisfy also a (j, k)-symmetry condition. As application of these results there is given the solution of a Fekete–Szegö type problem for a family of normalized biholomorphic starlike mappings in $${\mathbb {C}}^{n}.$$ C n .

Fekete–Szegö problem for Bavrin’s functions and close-to-starlike mappings in $${\mathbb {C}}^{n}$$

2022

Anal.Math.Phys.

The paper is devoted to the study of a family of complex-valued holomorphic functions and a family of holomorphic mappings in $${\mathbb {C}}^{n}.$$ C n . More precisely, the article concerns a Bavrin’s family of functions defined on a bounded complete n-circular domain $${\mathcal {G}}$$ G of $${\mathbb {C}}^{n}$$ C n and a family of biholomorphic mappings on the Euclidean open unit ball in $${\mathbb {C}}^{n}.$$ C n . The presented results include some estimates of a combination of the Fréchet differentials at the point $$z=0,$$ z = 0 , of the first and second order for Bavrin’s functions, also of the second and third order for biholomorphic close-to-starlike mappings in $${\mathbb {C}}^{n},$$ C n , respectively. These bounds give a generalization of the Fekete–Szegö coefficients problem for holomorphic functions of a complex variable on the case of holomorphic functions and mappings of several variables.