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

Abstract

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) ≤ s2 ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms qB(R, r) = 16Rr − 5r2 and QB(R, r) = 4R2 + 4Rr + 3r2; strongest in the sense that if Q is a quadratic form and s2 ≤ Q(R, r) ≤ QB(R, r) for all triangles then Q(R, r) = QB(R, r), and similarly for qB(R, r). In this paper we prove that QB (resp. qB) 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) ≤ s2 ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.