Proceedings of Symposia in Pure Mathematics
Volume 57 (1995)
An Improvement of the Osserman Constant
for the Bass Note of a Drum
RODRIGO BANUELOS AND TOM CARROLL
The following result of Hayman  answered an important question in the
study of vibrating membranes raised by Polya and Szego in [15, p. 16]:
(Hayman). Let D be a simply connected domain in the plane.
Let RD be the radius of the largest disc contained in D, and let XD be the
first Dirichlet eigenvalue for the Laplacian in D. There is a universal constant
a such that
Thus for a drum to produce an arbitrarily low tone it must necessarily
contain an arbitrarily large circular drum. The domain monotonicity of the
eigenvalue, which follows either from the variational formula or the proba-
bilistic characterization of kD (see below), implies that
where j0 is the smallest positive zero of the first Bessel function. The in-
equality (0.2) is sharp with equality attained when D is a disc. The Hayman
inequality was of great interest to analysts, geometers, and probabilists when
it appeared in 1976, and there have been many efforts to find the best con-
stant a and to identify the extremal domain. Hayman's original proof gave
a = 1/900. The references [1, 4, 5, 8, 14, 16], contain various proofs and
extensions of Hayman's theorem with various values of a . In  Osserman
proves the inequality with a = 1/4 and suggests that this may be a candidate
for the best constant. In Banuelos and Carroll , we prove that a 0.6197 .
In this note we present a different and simpler proof that a 0.325 , which
1991 Mathematics Subject Classification. Primary 31A35, 60J65.
The first author was supported in part by National Science Foundation grant 8958449-DMS.
©1995 American Mathematical Society
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