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 Certain integrals for classes of p-valent meromorphic functions

1982 ◽  
Vol 25 (1) ◽  
pp. 85-97 ◽  
Author(s):  
Vinod Kumar ◽  
S.L. Shukla
Keyword(s):  
Meromorphic Functions ◽  
Integral Operators ◽  
Starlike Functions ◽  
Image Position ◽  
Almost All

In this paper we introduce two classes, namely Γp(m, M) and Σp(m, M), of functionsregular and p-valent in D − {0} where D = {z : |z| < 1}. We show that, for suitable choices of real constants α, β and γ, the integral operators of the formmap into , where is the class of p-valent meromorphically starlike functions of order ρ, 0 ≤ ρ < 1. For the classes Γp(m, M) and Σp(m, M), we obtain class preserving integral operators of the formwith suitable restrictions on real constants α and γ.Our results generalize almost all known results obtained so far in this direction.

scholarly journals On a New Class ofp-Valent Meromorphic Functions Defined in Conic Domains

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Mohammed Ali Alamri ◽  
Maslina Darus
Keyword(s):  
Hypergeometric Function ◽  
Meromorphic Functions ◽  
Starlike Functions ◽  
Closure Properties ◽  
New Class ◽  
Conic Domain ◽  
Class Of Functions

We define a new class of multivalent meromorphic functions using the generalised hypergeometric function. We derived this class related to conic domain. It is also shown that this new class of functions, under certain conditions, becomes a class of starlike functions. Some results on inclusion and closure properties are also derived.

scholarly journals On a class of functions unifying the classes of Paatero, Robertson and others

1988 ◽  
Vol 11 (2) ◽  
pp. 251-258 ◽  
Author(s):  
S. Bhargava ◽  
S. Nanjunda Rao
Keyword(s):  
Variational Principle ◽  
Schwarzian Derivative ◽  
Starlike Functions ◽  
Class A ◽  
Boundary Rotation ◽  
Alpha Beta ◽  
Spirallike Functions ◽  
Convex And Starlike Functions ◽  
Number Of Classes ◽  
Class Of Functions

We study a classMkλ(α,β,b,c)of analytic functions which unifies a number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of orderβ, spirallike functions of orderβand functions for whichzf′is spirallike of orderβ, functions of boundary rotation utmostkπ,α-convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the classVkof functions of bounded boundary rotation, yield some representation theorems and a variational principle for our class. A consequence of these basic theorems is a theorem for this classMkλ(α,β,b,c)which unifies some earlier results concerning the radii of convexity of functions in the classVkλ(β)of Moulis and those concerning the radii of starlikeness of functions in the classesUkof Pinchuk andU2(β)of Robertson etc. By applying an estimate of Moulis concerning functions inVkλ(0), we obtain an inequality in the classMkλ(α,β,b,c)which will contain an estimate for the Schwarzian derivative of functions in the classVkλ(β)and in particular the estimate of Moulis for the Schwarzian of functions inVkλ(0).

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

scholarly journals On starlike functions

1980 ◽  
Vol 22 (2) ◽  
pp. 241-247
Author(s):  
V.P. Gupta ◽  
Iqbal Ahmad
Keyword(s):  
Starlike Functions ◽  
Open Disc ◽  
Image Position ◽  
Class Of Functions

Let S denote the class of functions f analytic and univalent in the open disc {z: |z| < 1} and normalized by f(0) = 0 = f′(0) − 1, and S*(α) denote the set of starlike functions of order α (0 ≤ α ≤ 1) in S. In this paper, the results of William M. Causey and William L. White [J. Math. Anal. Appl. 64 (1978), 458–466] are generalized by the following:THEOREM 1. Let f, g and h belong to S*(α), S*(λ) and S*(γ) , respectively and let F be defined bywhere a, c ∈ N. Then F belongs to S*(β) for |z| less than a suitably chosen number.THEOREM 2. Let F, g and h belong to S*(α), S*(λ) and S*(γ) , respectively and f be given by **. Then f belongs to S*(β) , for |z| less than a suitably chosen number.

Applications of the theory of cluster sets to a class of meromorphic functions

1957 ◽  
Vol 53 (1) ◽  
pp. 93-105 ◽  
Author(s):  
E. F. Collingwood ◽  
A. J. Lohwater
Keyword(s):  
Characteristic Function ◽  
Complex Number ◽  
Meromorphic Functions ◽  
Arbitrary Complex Number ◽  
Open Set ◽  
A Value ◽  
Arbitrary Complex ◽  
Cluster Sets ◽  
Image Position ◽  
Class Of Functions

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined bywhere m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned byfrom which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.

scholarly journals Coefficient estimates for starlike functions

1977 ◽  
Vol 16 (3) ◽  
pp. 415-425 ◽  
Author(s):  
M.L. Mogra ◽  
O.P. Juneja
Keyword(s):  
Unit Disc ◽  
Starlike Functions ◽  
Coefficient Estimates ◽  
Michigan Math ◽  
Image Position ◽  
Class Of Functions

Let (α β) denote the class of functionsanalytic in the unit disc Δ ≡{z: |z| < 1} and satisfyingfor some α, β (0 ≤ α < 1, 0 < β ≤ 1) and for all z ∈ Δ. In the present paper, sharp coefficient estimates for functions in (α, β) have been obtained. The results thus obtained not only generalize the corresponding results of Thomas H. MacGregor (Michigan Math. J. 10 (1963), 277–281), A.V. Boyd (Proc. Amer. Math. Soc. 17 (1966), 1016–1018) and others, but also give rise to analogous results for various other subclasses of starlike functions.

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