Coefficient inequalities for a new class of univalent functions

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:

• (i) f ∈ Σ _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 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.