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Infinitely many Gerretsen-Blundon style quadratic inequalities, all strongest in Blundon’s sense

机译:无限多的Gerretsen-Blundon风格的二次不等式,在Blundon的意义上都是最严重的

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摘要

Blundon has proved that if R, r and s are respectively the circumradius, the inradius and the semiperimeter of a triangle, then the strongest possible inequalities of the form q(R, r) ≤ s 2 ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms q B (R, r) = 16Rr ? 5r 2 and Q B (R, r) = 4R 2 + 4Rr + 3r 2; strongest in the sense that if Q is a quadratic form and s 2 ≤ Q(R, r) ≤ Q B (R, r) for all triangles then Q(R, r) = Q B (R, r), and similarly for q B (R, r). In this paper we prove that Q B (resp. q B ) is just one of infinitely many forms that appear as minimal (resp. maximal) elements in the partial order induced by the comparability relation in a certain set of forms, and we conclude that all these minimal forms are strongest in Blundon’s sense. We actually find all possible such strongest forms. Moreover we find all possible quadratic forms q, Q for which q(R, r) ≤ s 2 ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.
机译:Blundon证明,如果R,r和s分别是三角形的外接圆,半径和半周长,则q(R,r)≤s 2 ≤Q(R, r)对所有三角形成立,对等边的等式,其中q,Q为实二次型,对于Gerretsen形式q B (R,r)= 16Rr? 5r 2 和Q B (R,r)= 4R 2 + 4Rr + 3r 2 ;对于所有三角形,如果Q是二次形式且s 2 ≤Q(R,r)≤QB (R,r),则最强,则Q(R,r)= QB (R,r),并且对于q B (R,r)同样。在本文中,我们证明QB (分别是q B )只是在某种程度上由可比性关系引起的,以偏序的最小(最大)元素形式出现的无数形式之一。集合,我们得出结论,所有这些最小形式在Blundon的意义上都是最强的。实际上,我们找到了所有可能的最强形式。此外,我们找到所有可能的二次形式q,Q,其中所有三角形的q(R,r)≤s 2 ≤Q(R,r),并且对等边等式成立。

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