Schur’s inequality

If a, b, and c are non-negative real numbers and k1 is real, then the following inequality holds:


We can assume without loss of generality that cba via a permutationMathworldPlanetmath of the variables (as both sides are symmetricPlanetmathPlanetmath in those variables). Then collecting terms, we wish to show that


which is clearly true as every term on the left is positive.∎

There are a couple of special cases worth noting:

  • Taking k=1, we get the well-known

  • If c=0, we get (a-b)(ak+1-bk+1)0.

  • If b=c=0, we get ak+20.

  • If b=c, we get ak(a-c)20.

Title Schur’s inequality
Canonical name SchursInequality
Date of creation 2013-03-22 13:19:30
Last modified on 2013-03-22 13:19:30
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 11
Author rspuzio (6075)
Entry type Theorem
Classification msc 26D15