### Abstract

In the present article, a new class ∑ _{α} , 0 ≤ α < 1, of analytic and univalent functions f: U → C where U is an open unit disk, satisfying the standard normalization f(0) = f′(0) - 1 = 0 is considered. Assume that f ∑ _{α} takes the form such that A _{0,0} = 0 and A _{1,0} = 1. Also, we define the family Co(p), where p (0, 1), of functions f: U →C that satisfy the following conditions: (i) f ∑ _{α} is meromorphic in U and has a simple pole at the point p. (ii) f(0) = f′(0) - 1 = 0. (iii) f maps U conformally onto a set whose complement with respect toC is convex. We call such functions concave univalent functions. We prove some coefficient estimates for functions in this class when f has the expansion The second part of the article concerns some properties of a generalized Sǎlǎgean operator for functions in ∑ _{α} . Moreover, a result on subordination for the functions f ∑ _{α} is given.

Original language | English |
---|---|

Pages (from-to) | 221-229 |

Number of pages | 9 |

Journal | Lobachevskii Journal of Mathematics |

Volume | 29 |

Issue number | 4 |

DOIs | |

Publication status | Published - Oct 2008 |

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### Keywords

- Concave functions
- Convex set
- Meromorphic univalent functions
- Sǎlǎgean operator

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Lobachevskii Journal of Mathematics*,

*29*(4), 221-229. https://doi.org/10.1134/S1995080208040045

**Coefficient inequalities for a new class of univalent functions.** / Darus, Maslina; Ibrahim, R. W.

Research output: Contribution to journal › Article

*Lobachevskii Journal of Mathematics*, vol. 29, no. 4, pp. 221-229. https://doi.org/10.1134/S1995080208040045

}

TY - JOUR

T1 - Coefficient inequalities for a new class of univalent functions

AU - Darus, Maslina

AU - Ibrahim, R. W.

PY - 2008/10

Y1 - 2008/10

N2 - In the present article, a new class ∑ α , 0 ≤ α < 1, of analytic and univalent functions f: U → C where U is an open unit disk, satisfying the standard normalization f(0) = f′(0) - 1 = 0 is considered. Assume that f ∑ α takes the form such that A 0,0 = 0 and A 1,0 = 1. Also, we define the family Co(p), where p (0, 1), of functions f: U →C that satisfy the following conditions: (i) f ∑ α is meromorphic in U and has a simple pole at the point p. (ii) f(0) = f′(0) - 1 = 0. (iii) f maps U conformally onto a set whose complement with respect toC is convex. We call such functions concave univalent functions. We prove some coefficient estimates for functions in this class when f has the expansion The second part of the article concerns some properties of a generalized Sǎlǎgean operator for functions in ∑ α . Moreover, a result on subordination for the functions f ∑ α is given.

AB - In the present article, a new class ∑ α , 0 ≤ α < 1, of analytic and univalent functions f: U → C where U is an open unit disk, satisfying the standard normalization f(0) = f′(0) - 1 = 0 is considered. Assume that f ∑ α takes the form such that A 0,0 = 0 and A 1,0 = 1. Also, we define the family Co(p), where p (0, 1), of functions f: U →C that satisfy the following conditions: (i) f ∑ α is meromorphic in U and has a simple pole at the point p. (ii) f(0) = f′(0) - 1 = 0. (iii) f maps U conformally onto a set whose complement with respect toC is convex. We call such functions concave univalent functions. We prove some coefficient estimates for functions in this class when f has the expansion The second part of the article concerns some properties of a generalized Sǎlǎgean operator for functions in ∑ α . Moreover, a result on subordination for the functions f ∑ α is given.

KW - Concave functions

KW - Convex set

KW - Meromorphic univalent functions

KW - Sǎlǎgean operator

UR - http://www.scopus.com/inward/record.url?scp=57149125854&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57149125854&partnerID=8YFLogxK

U2 - 10.1134/S1995080208040045

DO - 10.1134/S1995080208040045

M3 - Article

AN - SCOPUS:57149125854

VL - 29

SP - 221

EP - 229

JO - Lobachevskii Journal of Mathematics

JF - Lobachevskii Journal of Mathematics

SN - 1995-0802

IS - 4

ER -