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On a Coefficient Body for Concave Functions

On a Coefficient Body for Concave Functions For $$p \in (0,1),$$ p ∈ ( 0 , 1 ) , let $$\mathcal C o_p$$ C o p be the class of meromorphic and univalent functions $$f$$ f in the unit disk $$\mathbb{D }$$ D with a simple pole at $$p$$ p such that $$\mathbb{C }\backslash f(\mathbb{D })$$ C \ f ( D ) is convex. These so-called concave functions can be expanded as $$\begin{aligned} f(z)= \sum _{n=0}^{\infty } a_n(f)z^n, \quad |z|<p \end{aligned}$$ f ( z ) = ∑ n = 0 ∞ a n ( f ) z n , | z | < p or $$\begin{aligned} f(z)= \sum _{n=-1}^{\infty } c_n(f) (z-p)^n, \quad |z-p|<1-p. \end{aligned}$$ f ( z ) = ∑ n = - 1 ∞ c n ( f ) ( z - p ) n , | z - p | < 1 - p . The present article shows a representation formula for functions of class $$\mathcal C o_p$$ C o p , using functions of positive real part, and gives an explicit description of the coefficient body $$\begin{aligned} \left\{ a_1(f), c_{-1}(f), c_1(f) \right\} . \end{aligned}$$ a 1 ( f ) , c - 1 ( f ) , c 1 ( f ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

On a Coefficient Body for Concave Functions

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References (9)

Publisher
Springer Journals
Copyright
Copyright © 2013 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-013-0018-y
Publisher site
See Article on Publisher Site

Abstract

For $$p \in (0,1),$$ p ∈ ( 0 , 1 ) , let $$\mathcal C o_p$$ C o p be the class of meromorphic and univalent functions $$f$$ f in the unit disk $$\mathbb{D }$$ D with a simple pole at $$p$$ p such that $$\mathbb{C }\backslash f(\mathbb{D })$$ C \ f ( D ) is convex. These so-called concave functions can be expanded as $$\begin{aligned} f(z)= \sum _{n=0}^{\infty } a_n(f)z^n, \quad |z|<p \end{aligned}$$ f ( z ) = ∑ n = 0 ∞ a n ( f ) z n , | z | < p or $$\begin{aligned} f(z)= \sum _{n=-1}^{\infty } c_n(f) (z-p)^n, \quad |z-p|<1-p. \end{aligned}$$ f ( z ) = ∑ n = - 1 ∞ c n ( f ) ( z - p ) n , | z - p | < 1 - p . The present article shows a representation formula for functions of class $$\mathcal C o_p$$ C o p , using functions of positive real part, and gives an explicit description of the coefficient body $$\begin{aligned} \left\{ a_1(f), c_{-1}(f), c_1(f) \right\} . \end{aligned}$$ a 1 ( f ) , c - 1 ( f ) , c 1 ( f ) .

Journal

Computational Methods and Function Theory – Springer Journals

Published: Jul 2, 2013

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