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.

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.

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

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.

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

On Density of Fourier Coefficients

1973 ◽  
Vol 16 (1) ◽  
pp. 93-103 ◽  
Author(s):  
Rafat N. Siddiqi
Keyword(s):  
Fourier Series ◽  
Total Variation ◽  
Bounded Variation ◽  
Fourier Coefficients ◽  
Function Of Bounded Variation ◽  
Fourier Sine ◽  
Image Position

Letfbe anLintegrable real valued function of period 2π and let(1)be its Fourier series. It is known that iffis of bounded variation then allnanandnbn(n=1,2,3,…) lie in the interval [-V(F)/π, V(F)/π;] whereV(f) is the total variation off. M. Izumi and S. Izumi [3] have recently asserted the following theorem A about the density of the positive and negative Fourier sine coefficients of a function of bounded variation.

A note on the representation of functions by absolutely convergent fourier integrals

1948 ◽  
Vol 44 (1) ◽  
pp. 8-12 ◽  
Author(s):  
H. R. Pitt
Keyword(s):  
Bounded Variation ◽  
Functions Of Bounded Variation ◽  
Integrable Functions ◽  
Image Position ◽  
Class Of Functions ◽  
Fourier Integrals

1. We write L for the class of integrable functions in (− ∞, ∞), V for the class of functions of bounded variation, and define A, A to be the classes of functions F(x) which may be expressed in the formsrespectively.

Representation of Functions as Weierstrass- Transforms

1967 ◽  
Vol 10 (5) ◽  
pp. 711-722 ◽  
Author(s):  
H.P. Heinig
Keyword(s):  
Bounded Variation ◽  
Finite Interval ◽  
Function Of Bounded Variation ◽  
Stieltjes Transform ◽  
Image Position

The Weierstrass - respectively Weierstrass - Stieltjes transform of a function F(t) or μ(t) is defined by1.1and1.2for all x for which these integrals converge. In what follows we shall always assume that F(t) is Lebesgue integrable in every finite interval and that μ(t) is a function of bounded variation.

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 On certain classes ofp-Valent functions

1986 ◽  
Vol 9 (1) ◽  
pp. 55-64 ◽  
Author(s):  
M. K. Aouf
Keyword(s):  
Critical Points ◽  
Complex Number ◽  
Bounded Boundary Rotation ◽  
Boundary Rotation ◽  
Class Of Functions

LetVkλ(α,b,p)(k≥2,b≠0is any complex number,0≤α<pand|λ|<π/2) denote the class of functionsf(z)=zp+∑n=p+1∞anznanalytic inU={z:|z|<1}having(p−1)critical points inUand satisfyinglimr→1−sup∫02π|Re{eiλ[p+1b(1+zf″(z)f′(z)−p)]−αcosλ}p−α|dθ≤kπcosλ.In this paper we generalize both those functionsf(z)which arep-valent convex of orderα,0≤α<p, with bounded boundary rotation and thosep-valent functionsf(z)for whichzf′(z)/pisλ-spirallike of orderα,0≤α<p.

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