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.

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

Toeplitz matrices whose elements are coefficients of Bazilevič functions

2018 ◽  
Vol 16 (1) ◽  
pp. 1161-1169
Author(s):  
Varadharajan Radhika ◽  
Jay M. Jahangiri ◽  
Srikandan Sivasubramanian ◽  
Gangadharan Murugusundaramoorthy
Keyword(s):  
Convex Functions ◽  
Toeplitz Matrices ◽  
Upper Bounds ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Bazilevic Functions ◽  
Class Of Functions

AbstractWe consider the Toeplitz matrices whose elements are the coefficients of Bazilevič functions and obtain upper bounds for the first four determinants of these Toeplitz matrices. The results presented here are new and noble and the only prior compatible results are the recent publications by Thomas and Halim [1] for the classes of starlike and close-to-convex functions and Radhika et al. [2] for the class of functions with bounded boundary rotation.

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.

scholarly journals On quasi-convex functions and related topics

1987 ◽  
Vol 10 (2) ◽  
pp. 241-258 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Unit Disc ◽  
Arc Length ◽  
Basic Properties ◽  
Complete Study ◽  
Coefficient Problems ◽  
Class Of Functions

LetSbe the class of functionsfwhich are analytic and univalent in the unit discEwithf(0)=0,f′(0)=1. LetC,S*andKbe the classes of convex, starlike and close-to-convex functions respectively. The classC*of quasi-convex functions is defined as follows:Letfbe analytic inEandf(0),f′(0)=1. Thenf ϵ C*if and only if there exists ag ϵ Csuch that, forz ϵ ERe(zf′(z))′g′(z)>0.In this paper, an up-to-date complete study of the classC*is given. Its basic properties, its relationship with other subclasses ofS, coefficient problems, arc length problem and many other results are included in this study. Some related classes are also defined and studied in some detail.

scholarly journals On subclasses of close-to-convex functions of higher order

1992 ◽  
Vol 15 (2) ◽  
pp. 279-289 ◽  
Author(s):  
Khalida Inayat Noor
Keyword(s):  
Convex Functions ◽  
Analytic Functions ◽  
Higher Order ◽  
Arc Length ◽  
Covering Theorem ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation

The classesTk(ρ),0≤ρ<1,k≥2, of analytic functions, using the classVk(ρ)of functions of bounded boundary rotation, are defined and it is shown that the functions in these classes are close-to-convex of higher order. Covering theorem, arc-length result and some radii problems are solved. We also discuss some properties of the classVk(ρ)including distortion and coefficient results.

Coefficient Behavior of a Class of Meromorphic Functions

1975 ◽  
Vol 27 (5) ◽  
pp. 1157-1165
Author(s):  
J. W. Noonan
Keyword(s):  
Meromorphic Functions ◽  
Starlike Functions ◽  
Boundary Rotation ◽  
Image Position ◽  
Class Of Functions

With , denote by Λk the class of functions ƒ of the formwhich are analytic in and which map y onto the complement of a domain with boundary rotation at most . It is known [2] that ƒ ∈ Λk if and only if there exist regular starlike functions s1 and s2, withsuch that

scholarly journals Generalized Alpha-Close-to-Convex Functions

2009 ◽  
Vol 2009 ◽  
pp. 1-9
Author(s):  
K. Inayat Noor ◽  
Halit Orhan ◽  
Saima Mustafa
Keyword(s):  
Convex Functions ◽  
Coefficient Estimates ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation

We define the classesGβ(α,k,γ)as follows:f∈Gβ(α,k,γ)if and only if, forz∈E={z∈ℂ:|z|<1},|arg{(1-α2z2)f′(z)/e−iβϕ′(z)}|≤γπ/2,0<γ≤1;α∈[0,1];β∈(−π/2,π/2), whereϕis a function of bounded boundary rotation. Coefficient estimates, an inclusion result, arclength problem, and some other properties of these classes are studied.

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