Kodai Mathematical Journal

Domains of variability of Laurent coefficients and the convex hull for the family of concave univalent functions

Bappaditya Bhowmik, Saminathan Ponnusamy, and Karl-Joachim Wirths

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Abstract

Let D denote the open unit disc and let p $\in$ (0,1). We consider the family Co(p) of functions f : D$\overline{{\mathbf C}}$ that satisfy the following conditions:

(i) f is meromorphic in D and has a simple pole at the point p.

(ii) f(0) = f′(0) – 1 = 0.

(iii) f maps D conformally onto a set whose complement with respect to $\overline{{\mathbf C}}$ is convex.

We determine the exact domains of variability of some coefficients an (f) of the Laurent expansion

$f(z)=\sum_{n=-1}^{\infty} a_n(f)(z-p)^n,$ |z – p|<1 – p,

for f $\in$ Co(p) and certain values of p. Knowledge on these Laurent coefficients is used to disprove a conjecture of the third author on the closed convex hull of Co(p) for certain values of p.

Article information

Source
Kodai Math. J. Volume 30, Number 3 (2007), 385-393.

Dates
First available: 1 November 2007

Permanent link to this document
http://projecteuclid.org/euclid.kmj/1193924942

Digital Object Identifier
doi:10.2996/kmj/1193924942

Mathematical Reviews number (MathSciNet)
MR2372126

Zentralblatt MATH identifier
1148.30006

Citation

Bhowmik, Bappaditya; Ponnusamy, Saminathan; Wirths, Karl-Joachim. Domains of variability of Laurent coefficients and the convex hull for the family of concave univalent functions. Kodai Mathematical Journal 30 (2007), no. 3, 385--393. doi:10.2996/kmj/1193924942. http://projecteuclid.org/euclid.kmj/1193924942.


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