An Elementary Counterexample to a Coefficient Conjecture

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 expansion f(z)=z+∑k=2∞akzk,|z|<p, and satisfying the condition |(zf(z))-z(zf(z))′-1|<λfor allz∈D, for some λ , 0 < λ< 1 . We denote this class by Up(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 | a2| and obtain inequalities for the initial coefficients of the Laurent series of f∈ Up(λ) at its pole. In [8] it had been conjectured that for f∈ Up(λ) the inequalities |an|≤1pn-1∑k=0n-1(λp2)k,n≥2, are valid. We provide a counterexample to this conjecture for the case n= 3 .