Schur's inequality

In mathematics, Schur's inequality, named after Issai Schur, establishes that for all non-negative real numbers x, y, z and t,

with equality if and only if x = y = z or two of them are equal and the other is zero. When t is an even positive integer, the inequality holds for all real numbers x, y and z.

When , the following well-known special case can be derived:

ProofEdit

Since the inequality is symmetric in   we may assume without loss of generality that  . Then the inequality

 

clearly holds, since every term on the left-hand side of the inequality is non-negative. This rearranges to Schur's inequality.

ExtensionsEdit

A generalization of Schur's inequality is the following: Suppose a,b,c are positive real numbers. If the triples (a,b,c) and (x,y,z) are similarly sorted, then the following inequality holds:

 

In 2007, Romanian mathematician Valentin Vornicu showed that a yet further generalized form of Schur's inequality holds:

Consider  , where  , and either   or  . Let  , and let   be either convex or monotonic. Then,

 

The standard form of Schur's is the case of this inequality where x = a, y = b, z = c, k = 1, ƒ(m) = mr.[1]

Another possible extension states that if the non-negative real numbers   with and the positive real number t are such that x + v ≥ y + z then[2]

 

NotesEdit

  1. ^ Vornicu, Valentin; Olimpiada de Matematica... de la provocare la experienta; GIL Publishing House; Zalau, Romania.
  2. ^ Finta, Béla (2015). "A Schur Type Inequality for Five Variables". Procedia Technology. 19: 799–801. doi:10.1016/j.protcy.2015.02.114.