MathLabs

Problem 2

Let a,b,c a,b,c be positive real numbers with abc=1 abc=1. Prove that (a−1+1b)(b−1+1c)(c−1+1a)≤1(a-1+\frac1b)(b-1+\frac1c)(c-1+\frac1a)\le 1.
Step 3 of 4: Apply Schur
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(x−y+z)(x+y−z)(−x+y+z)≤xyz.(x-y+z)(x+y-z)(-x+y+z)\le xyz.
Detailed analysis

If a factor is nonpositive the result is immediate; otherwise this is the degree-three Schur inequality.