Coefficients of Symmetric Functions of Bounded Boundary Rotation

1974 ◽  
Vol 26 (6) ◽  
pp. 1351-1355 ◽  
Author(s):  
Ronald J. Leach
Keyword(s):  
Unit Disc ◽  
Symmetric Functions ◽  
The Family ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Image Position

Let denote the family of all functions of the formthat are analytic in the unit disc U, f′(z) ≠ 0 in U and f maps U onto a domain of boundary rotation at most . Recently Brannan, Clunie and Kirwan [2] and Aharonov and Friedland [1] have solved the problem of estimating |amp+1| for all , provided m = 1.

On Odd Functions of Bounded Boundary Rotation

1974 ◽  
Vol 26 (3) ◽  
pp. 551-564
Author(s):  
Ronald J. Leach
Keyword(s):  
Unit Disc ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Definition Of ◽  
Image Position ◽  
Class Of Functions

Let VK denote the class of functionsthat are analytic in the unit disc U, satisfy f′(z) ≠ 0 in U, and map U onto a domain with boundary rotation at most Kπ (for a definition of this concept, see [9]). V. Paatero [9] showed that f(z) ∊ VK if and only if1.1

Multivalent and Meromorphic Functions of Bounded Boundary Rotation

1975 ◽  
Vol 27 (1) ◽  
pp. 186-199 ◽  
Author(s):  
Ronald J. Leach
Keyword(s):  
Critical Points ◽  
Analytic Functions ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation

The classVk(p). We generalize the class Vk of analytic functions of bounded boundary rotation [8] by allowing critical points in the unit disc U.Definition. Let f(z) = aqzq + . . . (q 1) be analytic in U. Then f(z) belongs to the class Vk(p) if for r sufficiently close to 1,andWe note that (1.1) implies that / has precisely p — 1 critical points in U.

ON CONVEX COMBINATIONS OF CONVEX HARMONIC MAPPINGS

2017 ◽  
Vol 96 (2) ◽  
pp. 256-262 ◽  
Author(s):  
ÁLVARO FERRADA-SALAS ◽  
RODRIGO HERNÁNDEZ ◽  
MARÍA J. MARTÍN
Keyword(s):  
Convex Domain ◽  
Unit Disc ◽  
Harmonic Functions ◽  
Recent Literature ◽  
Harmonic Mappings ◽  
The Family ◽  
Image Position

The family ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ of orientation-preserving harmonic functions $f=h+\overline{g}$ in the unit disc $\mathbb{D}$ (normalised in the standard way) satisfying $$\begin{eqnarray}h^{\prime }(z)+g^{\prime }(z)=\frac{1}{(1+\unicode[STIX]{x1D706}z)(1+\overline{\unicode[STIX]{x1D706}}z)},\quad z\in \mathbb{D},\end{eqnarray}$$ for some $\unicode[STIX]{x1D706}\in \unicode[STIX]{x2202}\mathbb{D}$, along with their rotations, play an important role among those functions that are harmonic and orientation-preserving and map the unit disc onto a convex domain. The main theorem in this paper generalises results in recent literature by showing that convex combinations of functions in ${\mathcal{F}}_{\unicode[STIX]{x1D706}}$ are convex.

scholarly journals On Certain Classes of Meromorphic Functions Associated with Conic Domains

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Khalida Inayat Noor ◽  
Fiaz Amber
Keyword(s):  
Differential Operator ◽  
Integral Operator ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Rate Of Growth ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Principle Of Subordination ◽  
Inclusion Results ◽  
Uniformly Bounded

Making use of the concept ofk-uniformly bounded boundary rotation and Ruscheweyh differential operator, we introduce some new classes of meromorphic functions in the punctured unit disc. Convolution technique and principle of subordination are used to investigate these classes. Inclusion results, generalized Bernardi integral operator, and rate of growth of coefficients are studied. Some interesting consequences are also derived from the main results.

scholarly journals On functions of bounded boundary rotation I

1969 ◽  
Vol 16 (4) ◽  
pp. 339-347 ◽  
Author(s):  
D. A. Brannan
Keyword(s):  
Upper Bound ◽  
Convex Functions ◽  
Image Domain ◽  
Basic Properties ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Image Position ◽  
Class Of Functions

Let Vk denote the class of functionswhich map conformally onto an image domain ƒ(U) of boundary rotation at most kπ (see (7) for the definition and basic properties of the class kπ). In this note we discuss the valency of functions in Vk, and also their Maclaurin coefficients.In (8) it was shown that functions in Vk are close-to-convex in . Here we show that Vk is a subclass of the class K(α) of close-to-convex functions of order α (10) for , and we give an upper bound for the valency of functions in Vk for K>4.

Curvature and Radius of Curvature for Functions with Bounded Boundary Rotation

1973 ◽  
Vol 25 (5) ◽  
pp. 1015-1023 ◽  
Author(s):  
J. W. Noonan
Keyword(s):  
Bounded Variation ◽  
Radius Of Curvature ◽  
Function Of Bounded Variation ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Image Position ◽  
Class Of Functions

For k ≧ 2 denote by Vk the class of functions f regular in and having the representation(1.1)where μ is a real-valued function of bounded variation on [0, 2π] with(1.2)Vk is the class of functions with boundary rotation at most kπ.

Coefficients of Functions with Bounded Boundary Rotation

1969 ◽  
Vol 21 ◽  
pp. 1477-1482 ◽  
Author(s):  
M. S. Robertson
Keyword(s):  
Analytic Functions ◽  
Bieberbach Conjecture ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Convex In One Direction ◽  
Image Position

For fixed k ≧ 2, let Vk denote the class of normalized analytic functionssuch that z ∈ E = {z; |z| <1} are regular and have f′(0) = l,f′(z) ≠ 0, and1Let Sk be the subclass of Vk whose members f(z) are univalent in E. It was pointed out by Paatero (4) that Vk coincides with Sk whenever 2 ≦ k ≦ 4. Later Rényi (5) showed that in this case, f(z) ∈ Sk is also convex in one direction in E. In (6) I showed that the Bieberbach conjectureholds for functions convex in one direction.

scholarly journals Toeplitz Matrices Whose Elements Are the Coefficients of Functions with Bounded Boundary Rotation

2016 ◽  
Vol 2016 ◽  
pp. 1-4 ◽  
Author(s):  
V. Radhika ◽  
S. Sivasubramanian ◽  
G. Murugusundaramoorthy ◽  
Jay M. Jahangiri
Keyword(s):  
Unit Disk ◽  
Toeplitz Matrices ◽  
Open Unit ◽  
Open Unit Disk ◽  
Toeplitz Determinants ◽  
Sharp Bounds ◽  
The Family ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation

Let R denote the family of functions f(z)=z+∑n=2∞anzn of bounded boundary rotation so that Ref′(z)>0 in the open unit disk U={z:z<1}. We obtain sharp bounds for Toeplitz determinants whose elements are the coefficients of functions f∈R.

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