Spirallike Mappings Research Articles

Fekete–Szegö Problems for Spirallike Mappings and Close-to-Quasi-Convex Mappings on the Unit Ball of a Complex Banach Space

Fekete–Szegö Problems for Spirallike Mappings and Close-to-Quasi-Convex Mappings on the Unit Ball of a Complex Banach Space

Fekete and Szegö inequality for a subclass of almost spirallike mappings of type β and order α on the bounded starlike circular domain in ℂ n

Let S ^ ( α , β ) be the familiar class of almost spirallike functions of type β and order α in the unit disk (see Definition 1.1). In this paper, first, we prove that for a function f ( z ) = z + ∑ n = 2 ∞ a n z n in the class S ^ ( α , β ) , then | a 3 − λ a 2 2 | ≤ ( 1 − α ) cos ⁡ β max { 1 , | 1 − 4 ( 1 − α ) ( 1 − λ ) cos ⁡ β e i β | } , λ ∈ C . The above estimation is sharp. Second, we extend this result to the bounded starlike circular domain in C n and obtain the sharp estimates. The results presented here would provide extensions of those given by Xu et al. [The Fekete and Szegö problem on the bounded starlike circular domain in C n . Pure Appl Math Q. 2016;12:621–638] and Xiong [Sharp coefficients bounds for class of almost starlike mappings of order α in C n . J Math Inequalities. 2020;14:853–865]. Finally, a certain conjecture is also formulated.

The Fekete-Szego problem for spirallike mappings and non-linear resolvents in Banach spaces

"We study the FeketeSzego problem on the open unit ball of a complex Banach space. Namely, the FeketeSzego inequalities are proved for the class of spirallike mappings relative to an arbitrary strongly accretive operator, and some of its subclasses. Next, we consider families of non-linear resolvents for holomorphically accretive mappings vanishing at the origin. We solve the Fekete- Szego problem over these families."

Note on the Fekete–Szegö Problem for Spirallike Mappings in Banach Spaces

In this note we present a remark on the paper “On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables” by Y. Lai and Q. Xu [10] published recently in the journal Results in Mathematics. We show that one of the theorems in [10] concerning the finite-dimensional space \({{\mathbb {C}}}^n\) is a direct consequence of another one, so it does not need an independent proof. Moreover, we prove that a sharp norm estimate on the Fekete–Szegö functional over spirallike mappings in a general Banach space can be deduced from a result in [10].

On the Coefficient Inequalities for a Class of Holomorphic Mappings Associated with Spirallike Mappings in Several Complex Variables

On the Coefficient Inequalities for a Class of Holomorphic Mappings Associated with Spirallike Mappings in Several Complex Variables

Properties of New Holomorphic Mappings with Respect to Conic Domains

This paper is mainly about holomorphic mappings associated with conic regions which are closely connected with k — ST(α). We introduce new subclasses of starlike (spirallike) functions, namely, S (k,α)(S (k,α,β)), and discuss their coefficient estimates and the Fekete-Szego-Goluzin’s problem. Then we generalize S (k,α,β) on the unit ball Bn in ℂn, that is, k-conic spirallike mappings of type β and order α. We obtain the growth, covering and distortion theorems of the generalized mappings. Besides that, we construct k-conic spirallike mappings of type β and order α on Bn through Sc(k,α,β) by the generalized Roper-Suffridge extension operators.

The Extension Operators on Bn+1 and Bounded Complete Reinhardt Domains

In this article, we extend the well-known Roper-Suffridge operator on Bn+1 and bounded complete Reinhardt domains in ℂn+1, then we investigate the properties of the generalized operators. Applying the Loewner theory, we obtain the mappings constructed by the generalized operators that have parametric representation on Bn+1. In addition, by using the geometric characteristics and the parametric representation of subclasses of spirallike mappings, we conclude that the extended operators preserve the geometric properties of several subclasses of spirallike mappings on Bn+1 and bounded complete Reinhardt domains in ℂn+1. The conclusions provide new approaches to construct mappings with special geometric properties in ℂn+1.

The Invariance of Subclasses of Biholomorphic Mappings on Bergman-Hartogs Domains

We mainly discuss the invariance of some subclasses of biholomorphic mappings under the generalized Roper-Suffridge operators on Bergman-Hartogs domains which are based on the unit ball Bn. Using the geometric properties and the distortion results of subclasses of biholomorphic mappings, we obtain the geometric characters of almost spirallike mappings of type β and order $$\alpha, S_\Omega^*(\beta, A, B)$$ , strong and almost spirallike mappings of type β and order α maintained under the generalized Roper-Suffridge operators on Bergman-Hartogs domains. Sequentially, we conclude that the generalized operators and the known operators preserve the same properties under some conditions. The conclusions generalize some known results.

Biholomorphic Mappings and the Extension Operators on New Hartogs Domains

In this paper, we generalize the Roper-Suffridge operator on the extended Hartogs domains. By using the geometric properties and the growth theorems of subclasses of biholomorphic mappings, we obtain the generalized operators preserve the properties of parabolic and spirallike mappings of type β and order ρ, $$S_\Omega^*(\beta, A, B)$$ , almost starlike mapping of complex order λ on ΩN under different conditions, and thus we get the corresponding results on the unit ball Bn in ℂn. The conclusions lead to some known results.

Subclasses of Biholomorphic Mappings Under the Extension Operators

In this article, we mainly study the invariance of some biholomorphic mappings with special geometric characteristics under the extension operators. First we generalize the Roper-Suffridge extension operators on Bergman-Hartogs domains. Then, by the geometric characteristics of subclasses of biholomorphic mappings, we conclude that the modified Roper-Suffridge operators preserve the properties of SΩ* (β,A,B), parabolic and spirallike mappings of type β and order ρ, strong and almost spirallike mappings of type β and order α as well as almost starlike mappings of complex order λ on $$\Omega_{p_{1}}^{B^{n}},\ldots,_{p_{s},q}$$ under different conditions, respectively. The conclusions provide new approaches to construct these biholomorphic mappings in several complex variables.

Almost spirallike mappings of type and complex order on the unit ball

ABSTRACTFirstly, we give a characterization of almost spirallike mappings of type and complex order on the unit ball in in terms of Loewner chains. Secondly, the growth and covering theorems for almost spirallike mappings of type and complex order on the unit ball are established by Loewner theory. In particular, some well-known results can be obtained using the conclusions in this article.

The generalized Roper-Suffridge operator on the unit ball in complex Banach and Hilbert spaces

In this paper, we extend the Roper-Suffridge extension operator in complex Banach space, and prove that the extended Roper-Suffridge operators preserve the properties of the subclasses of spirallike mappings on the unit ball in complex Banach spaces. Thereby, we promote the conclusions to the cases in complex Hilbert spaces. The conclusions provide new approaches to construct these subclasses of spirallike mappings in several complex variables.

The Roles Played by Order of Starlikeness and the Bloch Condition in the Roper–Suffridge Extension Operator

In this article, we obtain that the relation \(\Phi _{G,p}(f)({\mathcal {S}}_1^{*}(\alpha ))\subseteq {\mathcal {S}}_n^{*}\) implies G is a homogeneous polynomial of degree at most p for all \(\alpha \in [0, 1)\) on the bounded convex circular domain \(\Omega _{2,p}\). We shall also show that if f is a spirallike function of type \(\beta \) and is also a Bloch function on the unit disk, then \(\Phi _{G,p}(f)\) is a spirallike mapping of type \(\beta \) on the domain \(\Omega _{2,p}\) for certain G having terms of degree greater than p, where \(p\in {\mathbb {Z}}^+\) and \(p\ge 2\). As corollaries, some well known results can be got.

Geometric Mappings under the Perturbed Extension Operators in Complex Systems Analysis

In this paper, we mainly seek conditions on which the geometric properties of subclasses of biholomorphic mappings remain unchanged under the perturbed Roper-Suffridge extension operators. Firstly we generalize the Roper-Suffridge operator on Bergman-Hartogs domains. Secondly, applying the analytical characteristics and growth results of subclasses of biholomorphic mappings, we conclude that the generalized Roper-Suffridge operators preserve the geometric properties of strong and almost spiral-like mappings of typeβand orderα,SΩ⁎(β,A,B)as well as almost spiral-like mappings of typeβand orderαunder different conditions on Bergman-Hartogs domains. Sequentially we obtain the conclusions on the unit ballBnand for some special cases. The conclusions include and promote some known results and provide new approaches to construct biholomorphic mappings which have special geometric characteristics in several complex variables.

New subclasses of biholomorphic mappings and the modified Roper-Suffridge operator

The authors propose a new approach to construct subclasses of biholomorphic mappings with special geometric properties in several complex variables. The Roper-Suffridge operator on the unit ball Bn in Cn is modified. By the analytical characteristics and the growth theorems of subclasses of spirallike mappings, it is proved that the modified Roper-Suffridge operator [ΦG,γ (f)](z) preserves the properties of SΩ*(A,B), as well as strong and almost spirallikeness of type β and order α on Bn. Thus, the mappings in SΩ*(A,B), as well as strong and almost spirallike mappings, can be constructed through the corresponding functions in one complex variable. The conclusions follow some special cases and contain the elementary results.

Growth Estimates for the Numerical Range of Holomorphic Mappings and Applications

The numerical range of holomorphic mappings arises in many aspects of non-linear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings. In the present paper, we establish lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, non-linear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness.

The invariance of strong and almost spirallike mappings of type β and order α

The invariance of strong and almost spirallike mappings of type β and order a is discussed in this paper. From the maximum modulus principle of holomorphic functions, we obtain that the generalized Roper-Suffridge operators preserve strong and almost spirallikeness of type β and order α on the unit ball Bn in ℂn and on bounded and complete Reinhardt domains. Therefore we obtain that the generalized Roper-Suffridge operators preserve strong spirallikeness of type β, strong and almost starlikeness of order α, strong starlikeness on the corresponding domains. Thus we can construct more subclasses of spirallike mappings in several complex variables.

Extreme Points, Support Points and $$g$$ g -Loewner Chains Associated with Roper–Suffridge and Pfaltzgraff–Suffridge Extension Operators

In this paper we are concerned with the generalized Pfaltzgraff–Suffridge extension operator \(\Psi _{n,\alpha }\), \(\alpha \ge 0\), that provides a way of extending a locally biholomorphic mapping \(f\in H(B^n)\) to a locally biholomorphic mapping \(F\in H(B^{n+1})\). We obtain a subordination preserving result under the operator \(\Psi _{n,\alpha }\) and we consider extreme and support points associated with this operator. In the end, we present some examples of \(g\)-starlike mappings, \(g\)-spirallike mappings of type \(\alpha \) and \(g\)-almost starlike mappings of order \(\alpha \) on \(B^n\) and we consider the preservation of these notions under certain Roper–Suffridge extension operators.

The growth theorem and distortion theorem of spirallike mappings on B p

In this paper, by using the parametric representation of spirallike mappings, we construct to obtain the growth theorem of spirallike mappings on Reinhardt domain Bp. Moreover, the distortion theorem of spirallike mappings is obtained.

Asymptotically Spirallike Mappings in Reflexive Complex Banach Spaces

In this paper we consider the notion of asymptotic spirallikeness in reflexive complex Banach spaces \(X\), and the connection with univalent subordination chains. Poreda initially introduced the notion of asymptotic starlikeness to characterize biholomorphic mappings on the unit polydisc in \(\mathbb{C }^{n}\) which have parametric representation in the sense of Loewner theory. The authors introduced the notions of \(A\)-asymptotic spirallikeness and \(A\)-parametric representation on the Euclidean unit ball of \(\mathbb{C }^{n}\), where \(A\in L(\mathbb{C }^{n})\) with \(m(A)>0\). They showed that these notions are equivalent whenever \(k_+(A)<2m(A)\). In this paper we prove that if \(k_+(A)<2m(A)\) and \(f\in S(B)\) has \(A\)-parametric representation, then \(f\) is also \(A\)-asymptotically spirallike on the unit ball \(B\) of \(X\). For the converse, we need the additional assumption that \(f\) is a smooth \(A\)-asymptotically spirallike mapping, except in the finite-dimensional case \(X=\mathbb{C }^{n}\) with an arbitrary norm. The notion of asymptotic spirallikeness involves differential equations and may be regarded as giving a geometric characterization of certain domains in \(X\). That is one of the motivations for considering this notion in the case of reflexive complex Banach spaces.