Schur's inequality

Issai Schur (1875 - 1941) was a Jewish mathematician, born in what is now Belarus who studied and worked most of his life in Germany. He died in Tel-Aviv, Israel, two years after emigrating from Germany.

Among many significant results that bear his name, there is a surprising inequality with an instructive one-line proof:

For non-negative real numbers $x,$ $y,$ $z$ and a positive number $t,$

$x^t(x-y)(x-z)+y^t(y-z)(y-x)+z^t(z-x)(z-y)\ge 0.$

The equality holds in two cases:

  1. $x=y=z,$ or
  2. one of them is $0,$ while the other two are equal.

Proof

Because of the symmetry of the left-hand side in the variables $x,$ $y,$ $,$ we may assume without loss of generality that $x\ge y\ge z.$ Rewrite the inequality as

$(x-y)[x^t(x-z)-y^t(y-z)]+z^t(z-x)(z-y)\ge 0$

because, under the assumption $x\ge y\ge z,$ $x^t\ge y^t$ and $x-z\ge y-z,$ such that the two summands on the left are both non-negative.

That the equality holds in the two abovementioned cases is obvious. That these are the only two cases when this happens is more involved.

Special cases

For $t=1,$ Schur's inequality can be rearranged into

$\begin{align} x^3+y^3+z^3+3xyz &\ge x^2(y+z)+y^2(z+x)+z^2(x+y)\\ &=xy(x+y)+yz(y+z)+zx(z+x). \end{align}$

For $t=2,$ Schur's inequality can be rearranged into

$\begin{align} x^4+y^4+z^4+xyz(x+y+z) &\ge x^3(y+z)+y^3(z+x)+z^3(x+y)\\ &=xy(x^2+y^2)+yz(y^2+z^2)+zx(z^2+x^2). \end{align}$

Generalization

The only place where the positiveness of $t$ has been used was in establishing the implication $x\ge y\Rightarrow x^t\ge y^t.$ But this is true for any monotone increasing function $f$ such that there is an immediate generalization:

$f(x)(x-y)(x-z)+f(y)(y-z)(y-x)+f(z)(z-x)(z-y)\ge 0.$

This is so natural and straightforward that the appearance of the exponents (say, taking $f(u)=e^u)\;$ is sometimes described as a red herring - an artificial distraction from the essence of the problem.

There are of course further generalizations.

Reference

  1. J. Michael Steele, The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities, MAA, 2004

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