An Elementary Counterexample to a Coefficient Conjecture

In this article, we consider the family of functions f meromorphic in unit disk $${\mathbb D}=\{z:\,|z| < 1\}$$ with a pole at point $$z=p$$ , Taylor expansion $$\begin{aligned} f(z)= z+\sum _{k=2}^{\infty } a_kz^k, \quad |z|<p, \end{aligned}$$ and satisfying condition \left| \left( \frac{z}{f(z)}\right) -z\left( '-1\right| <\lambda \qquad \text {for all }z\in {\mathbb D}, for some $$\lambda $$ $$0<\lambda 1$$ . We denote class by $$\mathcal {U}_p(\lambda )$$ shall prove representation theorem class. As consequences, get simple proof estimates $$|a_2|$$ obtain inequalities initial coefficients Laurent series $$f\in \mathcal its pole. [8] it had been conjectured that |a_n|\le \frac{1}{p^{n-1}}\sum _{k=0}^{n-1}(\lambda p^2)^k, n\ge 2, are valid. provide counterexample to conjecture case $$n=3$$