An application of hypergeometric functions to a construction in several complex variables

Długosz, Renata; Liczberski, Piotr

doi:10.1007/s11854-019-0012-z

Cited by 3 publications

(4 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This family for k = 2 corresponds to a well-known Sakaguchi family [27] of a complex variable functions; strictly speaking of functions univalent starlike with respect to two symmetric points. In the paper [5] it was shown that for…”

Section: Denoting We Obtainmentioning

confidence: 99%

“…By F(G, ℂ m ), m ∈ ℕ 1 , let us denote the space of all functions f ∶ G ⟶ ℂ m , by H G the space of all holomorphic functions f ∈ F(G, ℂ) and by H G (0), H G (1) the collection of all f ∈ H G , normalized by f (0) = 0, f (0) = 1, respectively. Let us recall that every function f ∈ H G has a unique power series expansion where Q f ,m ∶ ℂ n → ℂ, m ∈ ℕ 0 , are m-homogeneous polynomials determined uniquelly by the mth Frechèt differential D m f (0) of f at zero via the formula Many authors considered some problems connected with m-homogeneous polynomials in the power series expansion (1.1) of functions from different subfamilies of H G (see for instance [2,5,9,13,24]). In particular, in the case G ⊂ ℂ 2 Bavrin [2] gives the following sharp estimates…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Długosz

Liczberski

2021

Annali di Matematica

Self Cite

View full text Add to dashboard 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 .

show abstract

Section: Denoting We Obtainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Długosz

Liczberski

2021

Annali di Matematica

Self Cite

View full text Add to dashboard Cite

show abstract

“…Since f 0,k is (0, k)-symmetrical part of f and L f 0,k is a (0, k)-symmetrical part of L f then the condition L f 0,k (z) 0,k = L f 0,k (z) holds and by (4) there exists the function h ∈ C G such that…”

mentioning

confidence: 99%

“…Proof. There is enough to use the fact that N k G is a subset of M k G and estimates µ G (Q f, m ) ≤ 1, m ∈ N in the family M k G and the proof of the path -connectness is similar to the investigations in [4]. □…”

mentioning

confidence: 99%

Some propositonerties of a Bavrin’s family of holomorphic functions in Cn

Długosz

Liczberski

Sibelska

et al. 2022

Filomat

Self Cite

View full text Add to dashboard Cite

In the [1], [4], [3] and [2] there were examined the Bavrin?s families (of holomorphic functions on bounded complete n? circular domains G ?Cn) in which the Temljakov operator Lf was presented as a product of a holomorphic function h with a positive real part and the (0, k)?symmetrical part of the function f,(k ? 2 is a positive integer). In [17] there was investigated the family of the above mentioned type, where the operator LLf was presented as a product of the same function h ? CG and (0, 2)-symmetrical part of the operator Lf. These considerations can be completed by the case of the factorization LLf by the same function h and the (0, k)-symmetrical part of operator Lf. In this article we will discuss the above case. In particular, we will present some estimates of a generalization of the norm of m-homogeneous polynomials Qf,m in the expansion of function f and we will also give a few relations between the different Bavrin?s families of the above kind.

show abstract

Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂⁿ

Sima

Xiong

2023

Demonstratio Mathematica

View full text Add to dashboard Cite

Let S γ , A , B ∗ ( D ) {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) be the usual class of g g -starlike functions of complex order γ \gamma in the unit disk D = { ζ ∈ C : ∣ ζ ∣ < 1 } {\mathbb{D}}=\left\{\zeta \in {\mathbb{C}}:| \zeta | \lt 1\right\} , where g ( ζ ) = ( 1 + A ζ ) ∕ ( 1 + B ζ ) g\left(\zeta )=\left(1+A\zeta )/\left(1+B\zeta ) , with γ ∈ C \ { 0 } , − 1 ≤ A < B ≤ 1 , ζ ∈ D \gamma \left\in {\mathbb{C}}\backslash \left\{0\right\}\right,-1\le A\lt B\le 1,\zeta \in {\mathbb{D}} . First, we obtain the bounds of all the coefficients of homogeneous expansions for the functions f ∈ S γ , A , B ∗ ( D ) f\in {S}_{\gamma ,A,B}^{\ast }\left({\mathbb{D}}) when ζ = 0 \zeta =0 is a zero of order k + 1 k+1 of f ( ζ ) − ζ f\left(\zeta )-\zeta . Second, we generalize this result to several complex variables by considering the corresponding biholomorphic mappings defined in a bounded complete Reinhardt domain. These main theorems unify and extend many known results.

show abstract

An application of hypergeometric functions to a construction in several complex variables

Cited by 3 publications

References 9 publications

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

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

Some propositonerties of a Bavrin’s family of holomorphic functions in Cn

Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂⁿ

Contact Info

Product

Resources

About

An application of hypergeometric functions to a construction in several complex variables

Cited by 3 publications

References 9 publications

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

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

Some propositonerties of a Bavrin’s family of holomorphic functions in Cn

Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂ n

Contact Info

Product

Resources

About

Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂⁿ