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 0. Introduction The following result of Hayman [9] answered an important question in the study of vibrating membranes raised by Polya and Szego in [15, p. 16]: THEOREM (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 (0.1) ^DalR2D. 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 (0.2) iDhlR2D, 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 [14] Osserman proves the inequality with a = 1/4 and suggests that this may be a candidate for the best constant. In Banuelos and Carroll [2], 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 0082-0717/95 $1.00+ $.25 per page 3 http://dx.doi.org/10.1090/pspum/057/1335458

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