Disproof of a coefficient conjecture for meromorphic univalent functions

Abstract:

Let $\Sigma$ denote the class of functions $g(z) = z + {b_0} + {b_1}{z^{ - 1}} + \cdots$ analytic and univalent in $|z|> 1$ except for a simple pole at $\infty$. A well-known conjecture asserts that $|{b_n}| \leq 2/(n + 1)\qquad (n = 1,2, \ldots )$ with equality for $g(z) = {(1 + {z^{n + 1}})^{2/(n + 1)}}/z = z + 2{z^{ - n}}/(n + 1) + \cdots$. Although the conjecture is true for $n=1,2$ and certain subclasses of the class $\Sigma$, the general conjecture is known to be false for all odd $n\ge 3$ and $n=4$. In $\S 2$, we generalize a variational method of Goluzin and develop second-variational techniques. This enables us in $\S 3$ to construct explicit counterexamples to the conjecture for all $n > 4$. In fact, the conjectured extremal function does not even provide a local maximum for ${\text {Re}}\{ {b_n}\}$, $n > 4$.