On the Univalence of Poly-analytic Functions

Abdulhadi, Zayid; Hajj, Layan El

doi:10.1007/s40315-021-00378-5

Cited by 5 publications

(5 citation statements)

References 30 publications

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“…In , Hachadi and Youssfi [18] have studied several properties of polyanalytic reproducing kernels. In , Abdulhadi and Hajj [1] extensively studied univalency criteria, Landau’s theorem, arc-length problem, and the Bohr phenomenon problem for polyanalytic functions in .…”

Section: Bohr Theorem For Operator-valued Polyanalytic Functionsmentioning

confidence: 99%

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Bohr operator on operator-valued polyanalytic functions on simply connected domains

Allu

Halder²

2023

Can. Math. Bull.

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In this article, we study the Bohr operator for the operator-valued subordination class $S(f)$ consisting of holomorphic functions subordinate to f in the unit disk $\mathbb {D}:=\{z \in \mathbb {C}: |z|<1\}$ , where $f:\mathbb {D} \rightarrow \mathcal {B}(\mathcal {H})$ is holomorphic and $\mathcal {B}(\mathcal {H})$ is the algebra of bounded linear operators on a complex Hilbert space $\mathcal {H}$ . We establish several subordination results, which can be viewed as the analogs of a couple of interesting subordination results from scalar-valued settings. We also obtain a von Neumann-type inequality for the class of analytic self-mappings of the unit disk $\mathbb {D}$ which fix the origin. Furthermore, we extensively study Bohr inequalities for operator-valued polyanalytic functions in certain proper simply connected domains in $\mathbb {C}$ . We obtain Bohr radius for the operator-valued polyanalytic functions of the form $F(z)= \sum _{l=0}^{p-1} \overline {z}^l \, f_{l}(z) $ , where $f_{0}$ is subordinate to an operator-valued convex biholomorphic function, and operator-valued starlike biholomorphic function in the unit disk $\mathbb {D}$ .

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Section: Bohr Theorem For Operator-valued Polyanalytic Functionsmentioning

confidence: 99%

“…(1) For a particular case h(z) ≡ I, Theorem 2.2 reduces to Theorem 2.1. By taking f , g, h ∶ D → C are analytic functions in Theorem 2.2, we obtain the scalar-valued quasi-subordination result, which has been established in[5].…”

mentioning

confidence: 99%

Bohr operator on operator-valued polyanalytic functions on simply connected domains

Allu

Halder²

2023

Can. Math. Bull.

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“…It is well-known that every bi-analytic function f in a simply connected domain D has the representation (cf. [1])…”

Section: Definitions and Notationsmentioning

confidence: 99%

“…Laplacian equation ) in D , where we use the common notations for its formal derivatives: Note also that It is well-known that every bi-analytic function f in a simply connected domain D has the representation (cf. [1]) where g and h are complex-valued analytic functions in D . Similarly, every harmonic function f in a simply connected domain D can be written as with , where g and h are analytic on D (for details, see [11]).…”

Section: Introductionmentioning

confidence: 99%

Landau-type theorems for certain bounded bi-analytic functions and biharmonic mappings

Liu¹,

Ponnusamy²

2023

Can. Math. Bull.

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In this article, we establish three new versions of Landau-type theorems for bounded bi-analytic functions of the form $F(z)=\bar {z}G(z)+H(z)$ , where G and H are analytic in the unit disk with $G(0)=H(0)=0$ and $H'(0)=1$ . In particular, two of them are sharp, while the other one either generalizes or improves the corresponding result of Abdulhadi and Hajj. As consequences, several new sharp versions of Landau-type theorems for certain subclasses of bounded biharmonic mappings are proved.

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“…In 2019, Hachadi and Youssfi [18] have studied several properties of polyanalytic reproducing kernels. In 2021, Abdulhadi and Hajj [1] extensively studied univalency criteria, Landau's theorem, arc-length problem, and the Bohr phenomenon problem for polyanalytic functions in D.…”

Section: Bohr Theorem For Operator-valued Polyanalytic Functionsmentioning

confidence: 99%

Bohr operator on opertor valued polyanalytic functions on simply connected domains

Allu¹,

Halder²

2021

Preprint

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In this article, we study the Bohr operator for the operator valued subordination class S(f ) consisting of holomorphic functions subordinate to f in the unit disk D := {z ∈ C : |z| < 1}, where f : D → B(H) is holomorphic and B(H) is the algebra of bounded linear operators on a complex Hilbert space H. We establish several subordination results, which can be viewed as the analogues of a couple of interesting subordination results from scalar valued settings. We also obtain a von Neumann-type inequality for the class of self-analytic mappings of the unit disk D which fix the origin. Furthermore, we extensively study Bohr inequalities for operator valued polyanalytic functions in certain proper simply connected domains in C. We obtain Bohr radius for the operator valued polyanalytic functions of the form F (z) = p−1 l=0 z l f l (z), where f 0 is subordinate to an operator valued convex biholomorphic function, and operator valued starlike biholomorphic function in the unit disk D.

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On the Univalence of Poly-analytic Functions

Cited by 5 publications

References 30 publications

Bohr operator on operator-valued polyanalytic functions on simply connected domains

Bohr operator on operator-valued polyanalytic functions on simply connected domains

Landau-type theorems for certain bounded bi-analytic functions and biharmonic mappings

Bohr operator on opertor valued polyanalytic functions on simply connected domains

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