[59] (with P. R. Garabedian) A coefficient inequality for schlicht functions

Abstract

Let Σ ₀ denote the class of functions $g(z) = z +\sum _{ n=1}^{\infty }b_{n}{z}^{-n}$ analytic and univalent in |z| > 1. The area theorem says that ∑ _n=1 ^∞ n|b _n |² ≤ 1 so that $\vert b_{n}\vert \leq 1/\sqrt{n}$, but the latter inequality is sharp only for n = 1. In 1938, Schiffer (Bull. Soc. Math. France, 66, 48–55, 1938) proved that |b ₂|≤ $\frac{2}{3}$, with equality only for $g(z) = z{\left (1 + {z}^{-3}\right )}^{2/3}$ and its rotations. This result gave rise to the conjecture that $\vert b_{n}\vert \leq \frac{2} {n+1}$, with equality only for rotations of

$$\displaystyle{ g(z) = z{\left (1 + {z}^{-(n+1)}\right )}^{2/(n+1)} = z + \frac{2} {n + 1}{z}^{-n} +\ldots \, . }$$

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