An elementary counterexample to a coefficient conjecture

Abstract

In this article, we consider the family of functions $f$ meromorphic in the unit disk $\ID=\{z :\,|z| < 1\}$ with a pole at the point $z=p$, a Taylor expansion \[f(z)= z+\sum_{k=2}^{\infty} a_kz^k, \quad |z|<p, \] and satisfying the condition \[\left |\left(\frac{z}{f(z)}\right)-z\left(\frac{z}{f(z)}\right)'-1\right |<\lambda,\, \forall z\in\ID, \] for some $\lambda$, $0<\lambda < 1$. We denote this class by $\mathcal{U}_m(\lambda)$ 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 $f\in \mathcal{U}_m(\lambda)$ at its pole. In \cite{PW2} it had been conjectured that for $f\in \mathcal{U}_m(\lambda)$ the inequalities \[|a_n|\,\leq\,\frac{1}{p^{n-1}}\sum_{k=0}^{n-1}(\lambda p^2)^k, \quad n\geq 2 \] are valid. We provide a counterexample to this conjecture for the case $n=3$.