Lobachevskii Journal of Mathematics

, Volume 29, Issue 4, pp 221–229

# Coefficient inequalities for a new class of univalent functions

Article

## Abstract

In the present article, a new class Σ α , 0 ⩽ α < 1, of analytic and univalent functions f: U$$\bar C$$, where U is an open unit disk, satisfying the standard normalization f(0) = f′(0) − 1 = 0 is considered. Assume that f ∈ Σ α takes the form
$$f(z) = z + \sum\limits_{n = 2}^\infty {A_{n,\alpha } (f)z^{n + \alpha } } , 0 \leqslant \alpha < 1,$$
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$$\bar C$$ that satisfy the following conditions:
1. (i)

f ∈ Σ α is meromorphic in U and has a simple pole at the point p.

2. (ii)

f(0) = f′(0) − 1 = 0.

3. (iii)

f maps U conformally onto a set whose complement with respect to $$\bar C$$ is convex.

We call such functions concave univalent functions. We prove some coefficient estimates for functions in this class when f has the expansion
$$f(z) = \sum\limits_{n = 0}^\infty {a_{n,\alpha } (z - p)^{n + \alpha } } , \left| {z - p} \right| < 1 - p, p \in (0,1), 0 \leqslant \alpha < 1.$$
The second part of the article concerns some properties of a generalized Sălăgean operator for functions in Σ α . Moreover, a result on subordination for the functions f ∈ Σ α is given.

## Key words and phrases

meromorphic univalent functions concave functions convex set Sălăgean operator

30C45

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