## Kodai Mathematical Journal

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

### 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

#### Abstract

Let **D** denote the open unit disc and let *p* (0,1). We consider the family *Co*(*p*) of functions *f* : **D** → 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 is convex.

We determine the exact domains of variability of some coefficients *a*_{n} (*f*) of the Laurent expansion

|z – p|<1 – p,

for *f* *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 in Project Euclid: 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 Math. J. 30 (2007), no. 3, 385--393. doi:10.2996/kmj/1193924942. http://projecteuclid.org/euclid.kmj/1193924942.