On the representation formulas for the functions in class Σ*(𝑝,𝑤₀)

Chang, Yuh Lin

doi:10.1090/s0002-9939-1988-0943077-1

Cited by 2 publications

(3 citation statements)

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“…As a consequence of Theorem 3.1, we have the following result which has been proved in [11] but by using a different method of proof. Corollary 3.1.…”

Section: Now Writingmentioning

confidence: 81%

Coefficient inequalities for concave and meromorphically starlike univalent functions

Bhowmik

Ponnusamy

2008

Ann. Polon. Math.

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Abstract. Let D denote the open unit disk and f : D → C be meromorphic and univalent in D with a simple pole at p ∈ (0, 1) and satisfying the standard normalization f (0) = f ′ (0) − 1 = 0. Also, assume that f has the expansionand maps D onto a domain whose complement with respect to C is a convex set (starlike set with respect to a point w 0 ∈ C, w 0 = 0 resp.). We call such functions concave (meromorphically starlike resp.) univalent functions and denote this class by Co(p) (Σ s (p, w 0 ) resp.). We prove some coefficient estimates for functions in these classes; the sharpness of these estimates is also established.1. Introduction. One of the most interesting questions in the theory of univalent functions is to find the region of variability of the nth Taylor (Laurent resp.) coefficient for functions f that are analytic (meromorphic resp.) and univalent in the unit disk D = {z : |z| < 1}. The leading example is the Bieberbach conjecture settled by de Branges in 1985 for the class S of normalized analytic univalent functions f in D although corresponding results for important subclasses of S are relatively easy and were settled positively much earlier.In this paper, we consider the family Co(p) of functions f : D → C that satisfy the following conditions:

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“…As a consequence of Theorem 3.1, we have the following result which has been proved in [11] but by using a different method of proof. Corollary 3.1.…”

Section: Now Writingmentioning

confidence: 81%

Coefficient inequalities for concave and meromorphically starlike univalent functions

Bhowmik

Ponnusamy

2008

Ann. Polon. Math.

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“…The necessary part of Theorem A has been proved by Miller [9] while the sufficiency part has been established by Yuh Lin [6,Theorem 1]. In [8,9], Miller discussed a numbers of properties of the class Σ * (p, w 0 ).…”

Section: Introductionmentioning

confidence: 99%

“…In [8,9], Miller discussed a numbers of properties of the class Σ * (p, w 0 ). See also [3,6,11] for some other basic results such as bounds for |f (z) − w 0 |…”

Section: Introductionmentioning

confidence: 99%

On Some Problems of James Miller

Bhowmik

Ponnusamy

Wirths³

2010

Cubo

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We consider the class of meromorphic univalent functions having a simple pole at p ∈ (0, 1) and that map the unit disc onto the exterior of a domain which is starlike with respect to a point w0 = 0, ∞. We denote this class of functions by Σ * (p, w0). In this paper, we find the exact region of variability for the second Taylor coefficient for functions in Σ * (p, w0). In view of this result we rectify some results of James Miller. RESUMENConsideramos la clase de funciones univalentes meromoforficos teniendo un polo simple en p ∈ (0, 1) y la aplicación del disco unitario sobre el exterior de un dominio el cual es estrellado con respecto al punto w0 = 0, ∞. Denotamos esta clase de funciones por Σ * (p, w0). En este artículo encontramos la región exacta de variabilidad del segundo coeficiente de Taylor para funciones in Σ * (p, w0). En vista de estos resultados nosotros rectificamos algunos resultados de James Miller. 12, 1 (2010)

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On the representation formulas for the functions in class Σ*(𝑝,𝑤₀)

Abstract: ABSTRACT. We prove the converse of J. Miller's integral representation theorem for functions in the class E(p,u;o) of starlike meromorphic univalent functions. As an application we improve the bounds of the modulus. We also observe some properties of the class ST(P) = [J T,(p,u>o).

Cited by 2 publications

References 6 publications

Coefficient inequalities for concave and meromorphically starlike univalent functions

Coefficient inequalities for concave and meromorphically starlike univalent functions

On Some Problems of James Miller

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On the representation formulas for the functions in class Σ*(𝑝,𝑤₀)

Abstract: ABSTRACT. We prove the converse of J. Miller's integral representation theorem for functions in the class E*(p,u;o) of starlike meromorphic univalent functions. As an application we improve the bounds of the modulus. We also observe some properties of the class ST(P) = [J T,*(p,u>o).

Cited by 2 publications

References 6 publications

Coefficient inequalities for concave and meromorphically starlike univalent functions

Coefficient inequalities for concave and meromorphically starlike univalent functions

On Some Problems of James Miller

Contact Info

Product

Resources

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Abstract: ABSTRACT. We prove the converse of J. Miller's integral representation theorem for functions in the class E(p,u;o) of starlike meromorphic univalent functions. As an application we improve the bounds of the modulus. We also observe some properties of the class ST(P) = [J T,(p,u>o).