On the Taylor Coefficients of a Subclass of Meromorphic Univalent Functions

Abstract

Let V p (λ) be the collection of all functions f defined in the unit disc D having a simple pole at z = p where 0 < p < 1 and analytic in D \ {p} with f (0) = 0 = f ′ (0)−1 and satisfying the differential inequality |(z/f (z)) 2 f ′ (z)−1| < λ for z ∈ D, 0 < λ ≤ 1. Each f ∈ V p (λ) has the following Taylor expansion: f (z) = z + ∞ n=2

An elementary counterexample to a coefficient conjecture

“…As the title indicates, the main aim of this paper is to disprove the conjecture (4). We want to emphasize at this place that some of the other results can be found in [2,3,4]. As a byproduct of main concern, we include some easy consequences of our discussion in the form of corollaries.…”

“…For f ∈ U m (λ) with the expansion (1), in [8] it has been proved by an application of Banach's fixed point theorem that the sharp inequality (3) |a 2 | ≤ 1 + λp 2 p is valid, and it has been conjectured there that the inequalities (4)…”