Schur's inequality explained

In mathematics, Schur's inequality, named after Issai Schur,establishes that for all non-negative real numbersx, y, z, and t>0,

x ^{t} (x - y)(x - z) + y ^{t} (y - z)(y - x) + z ^{t} (z - x)(z - y) \ge 0

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

t =1 , the following well-known special case can be derived:x ^{3} + y ^{3} + z ^{3} + 3 xyz \geq xy(x + y) + xz(x + z) + yz(y + z)

Proof

Since the inequality is symmetric in

x,y,z we may assume without loss of generality that x \geq y \geq z . Then the inequality(x - y)[x ^{t(x-z)-y} ^{t(y-z)]+z} ^{t(x-z)(y-z)} \geq 0

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

Extensions

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:

a (x - y)(x - z) + b (y - z)(y - x) + c (z - x)(z - y) \ge 0 .

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

Consider

a,b,c,x,y,z \in R , where a \geq b \geq c , and either x \geq y \geq z or z \geq y \geq x . Let k \in Z ^{+} , and let f: R → R ^{+} _{0} be either convex or monotonic. Then,{f(x)(a - b) ^{k(a-c)} ^{k+f(y)(b-a)} ^{k(b-c)} ^{k+f(z)(c-a)} ^{k(c-b)} ^{k} \geq 0 }. The standard form of Schur's is the case of this inequality where x = a, y = b, z = c, k = 1, ƒ(m) = m^r.^[1]

Another possible extension states that if the non-negative real numbers

x \geq y \geq z \geq v with and the positive real number t are such that x + v ≥ y + z then^[2] x ^{t} (x - y)(x - z)(x - v) + y ^{t} (y - x)(y - z)(y - v) + z ^{t} (z - x)(z - y)(z - v) + v ^{t} (v - x)(v - y)(v - z) \ge 0 .

Notes and References

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