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In this paper, we establish a number of inequalities involving symmetric means and symmetric harmonic means. We then apply these new inequalities to obtain many geometric inequalities of isoperimetric type for plane polygons.
Let M be a smooth bounded domain in Rn with smooth boundary, n ≥ 2, and . We prove an inequality involving the first k + 1 eigenvalues of the eigenvalue problem: where am−1 ≥ 0 are constants and at−1 = 1. We also obtain a uniform estimate of the upper bound of the ratios of consecutive eigenvalues.
Throughout this paper, the Alexander-Spanier cohomology with compact supports will be used. Suppose X is a compact connected topological ra-manifold which admits an effective action of a compact connected Lie group G (m ≧ 19).
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