Areas Research Papers - Academia.edu (original) (raw)

The Erdős-Debrunner inequality referred to in the title states that “If a triangle XYZ is inscribed in a triangle ABC – with X,Y,Z on the sides BC,CA, and AB – then σ(XYZ)≥min(σ(BXZ),σ(CXY),σ(AYZ)) with equality if and only if X,Y, and Z... more

The Erdős-Debrunner inequality referred to in the title states that “If a triangle XYZ is inscribed in a triangle ABC – with X,Y,Z on the sides BC,CA, and AB – then σ(XYZ)≥min(σ(BXZ),σ(CXY),σ(AYZ)) with equality if and only if X,Y, and Z are the midpoints of the sides BC,CA, and AB,” where σ(MNP) stands for the area of the triangle MNP. In [Elem. Math. 61, No. 1, 32–35 (2006; Zbl 1135.51017)], W. Janous has generalized this inequality in the following manner: if by M p (x,y,z) we denote the mean (x p +y p +z p 3) 1/p for p≠0 and min(x,y,z) for p=-∞, then σ(XYZ)≥M p (σ(BXZ),σ(CXY),σ(AYZ))(1) holds for certain negative values of p. He also asked for the greatest value of p for which (1) holds and formulated two conjectures aimed at establishing that maximum value of p. The authors of this paper prove both conjectures by methods of calculus – different from those of V. Mascioni [JIPAM, J. Inequal. Pure Appl. Math. 8, No. 2, Paper No. 32, 5 p. (2007; Zbl 1134.51017)], who also settled t...