A new inequality for symmetric functions

✍ Scribed by Gustave A. Efroymson; Blair Swartz; Burton Wendroff

• Publisher: Elsevier Science
• Year: 1980
• Tongue: English
• Weight: 792 KB
• Volume: 38
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(80)90001-8

✦ Synopsis

Let x, > 0, y, ) 0 for i = I,..., n; and let a,(x) be the elementary symmetric function of n variables given by a,(x) = C ,<rl<...<,,<n~r,...~r,.Definethepartial ordering x< y if a,(x) (a,(y), j= l,..., n. We show that x<y=-x"<y", 0 ( a < 1, where (x"), = x7 . We also give a necessary and sufficient condition on a function f(t) such that x < y =+ f(x) <J(v). Both results depend crucially on the following: If x < y there exists a piecewise differentiable path z(t), with zi(t) ) 0, such that z(0) = x, r(l) = y, and z(s) <z(t) if 0 <s < t < 1.