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.

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.

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.

scholarly journals On a class of analytic functions

1975 ◽  
Vol 20 (1) ◽  
pp. 46-53
Author(s):  
R. M. Goel
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Image Position ◽  
Class Of Functions

Let S(α) denote the class of functions regular and analytic in the unit disc E = {z¦z¦< 1¦} and satisfying the condition, .

scholarly journals Univalent functions with univalent Gelfond-Leontev derivatives

1984 ◽  
Vol 29 (3) ◽  
pp. 329-348 ◽  
Author(s):  
O.P. Juneja ◽  
S.M. Shah
Keyword(s):  
Unit Disk ◽  
Shift Operator ◽  
Univalent Functions ◽  
Nondecreasing Sequence ◽  
Growth Properties ◽  
Linear Invariant ◽  
Definition Of ◽  
Image Position ◽  
Linear Invariant Family ◽  
Class Of Functions

Let be a nondecreasing sequence of positive numbers. We consider Gelfond-Leontev derivative Df(z), of a function , defined by for univalence and growth properties, and extend some results of Shah and Trimble. Set en = {d1d2 … dn), n≥l, e0 = 1, . Let r be the radius of convergence of p(z). We state parts of Theorem 1 and Corollaries. Let f and all Dkf, k = 1, 2,…, be analytic and univalent in the unit disk U. Then(iii) if p is entire and of growth (ρ, T) then f must be entire and of growth not exceeding (ρ, 2d2T),(iv) if D corresponds to the shift operator (dn ≡ l), then .Another class of functions is defined by a condition of the form |an+1/an| ≤ bn+1/dn+1, where is a sequence of positive numbers satisfying and inequality, and it is shown that all functions in this class along with all their Gelfond–Leontev successive derivatives are regular and univalent in U. An extension of the definition of a linear invariant family is given and results analogous to (i) and (ii) are stated.

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 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.

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.

Sign in / Sign up

Export Citation Format

Share Document