An elementary counterexample to a coefficient conjecture

Abstract

In this article, we consider the family of functions f meromorphic in the unit disk D = {z : |z| < 1} with a pole at the point z = p, a Taylor expansionand satisfying the conditionfor some λ, 0 < λ < 1. We denote this class by U m (λ) and we shall prove a representation theorem for the functions in this class. As consequences, we get a simple proof for the estimates of |a 2 | and obtain inequalities for the initial coefficients of the Laurent series of f ∈ U m (λ) at its pole. In [8] it had been conjectured that for f ∈ U m (λ) the inequalitiesare valid. We provide a counterexample to this conjecture for the case n = 3.