An elementary counterexample to a coefficient conjecture

In this article, we consider the family of functions $f$ meromorphic in the unit disk $\text{ID}=\{z:\mathrm{\mid }z\mathrm{\mid }<1\}$ with a pole at the point $z=p$, a Taylor expansion $$f(z)=z+\sum _{k=2}^{\mathrm{\infty }}{a}_k{z}^k,\mathrm{\mid }z\mathrm{\mid }<p,$$ and satisfying the condition $$\mid \left(\frac{z}{f(z)}\right)-z{\left(\frac{z}{f(z)}\right)}^{\mathrm{\prime }}-1\mid <\lambda ,\mathrm{\forall }z\in \text{ID},$$ 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 $\mathrm{\mid }{a}_2\mathrm{\mid }$ 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 $$\mathrm{\mid }{a}_n\mathrm{\mid }\le \frac{1}{p^{n-1}}\sum _{k=0}^{n-1}(\lambda p^2{)}^k,n\ge 2$$ are valid. We provide a counterexample to this conjecture for the case $n=3$.