By Catherine Bandle, Attila Gilányi, László Losonczi, Zsolt Páles, Michael Plum

Inequalities proceed to play an important function in arithmetic. possibly, they shape the final box comprehended and utilized by mathematicians in all parts of the self-discipline. because the seminal paintings Inequalities (1934) by way of Hardy, Littlewood and P?lya, mathematicians have laboured to increase and sharpen their classical inequalities. New inequalities are came upon each year, a few for his or her intrinsic curiosity while others move from effects bought in a number of branches of arithmetic. The research of inequalities displays the numerous and numerous facets of arithmetic. On one hand, there's the systematic look for the fundamental ideas and the research of inequalities for his or her personal sake. however, the topic is the resource of inventive principles and strategies that supply upward thrust to possible trouble-free yet however severe and tough difficulties. there are various purposes in a wide selection of fields, from mathematical physics to biology and economics.

This quantity comprises the contributions of the members of the convention on Inequalities and Applications held in Noszvaj (Hungary) in September 2007. it really is conceived within the spirit of the previous volumes of the overall Inequalities conferences held in Oberwolfach from 1976 to 1995 within the feel that it not just includes the most recent effects awarded by means of the contributors, however it can be an invaluable reference ebook for either academics and study employees. The contributions mirror the ramification of normal inequalities into many components of arithmetic and in addition current a synthesis of leads to either conception and practice.

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1. Introduction We study an eigenvalue problem with a spectral parameter in a boundary condition. This problem for the two-dimensional Laplace equation is relevant to sloshing frequencies that describe free oscillations of an inviscid, incompressible, heavy fluid in a canal having uniform cross-section and bounded from above by a horizontal free surface. It is demonstrated that accurate bounds for the smallest eigenvalues can be computed using inclusion theorems (based on variational principles) and interval arithmetic.

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It has been observed in [1] that the fact that the eigenfunctions ϕ1 and ϕ−1 corresponding to λ1 and to λ−1 are of constant sign depends on the size of σ. More precisely we have: (i) If σ < σ0 then ϕ1 is of constant sign and λ1 is simple, whereas ϕ−1 changes sign. (ii) If σ > σ0 then ϕ−1 is of constant sign and λ−1 is simple, whereas ϕ1 changes sign. (iii) If σ = σ0 both ϕ1 and ϕ−1 change sign. The main result of this paper is a Rayleigh-Faber-Krahn type inequality. Theorem 1. Let D∗ be the ball of the same volume as D.

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