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Let λ 1  and λ 2  (λ 1  &amp;lt; λ 2 ) be the two lowest eigenvalues of a membrane. We show that the inequality λ 2 /λ 1  &amp;lt; 2.6578 is always fulfilled, irrespective of the shape of the membrane. This is an improvement upon results due to BRANDS, PAYNE, PÓLYA and WEINBERGER.

On the Upper Bound for the Ratio of the First Two Membrane Eigenvalues