MathLabs

Problem 2

Let a,b,ca,b,c be positive real numbers with abc=1abc=1. Prove that 1a3(b+c)+1b3(c+a)+1c3(a+b)≥32\frac1{a^3(b+c)}+\frac1{b^3(c+a)}+\frac1{c^3(a+b)}\ge\frac32.
Step 2 of 3: Apply Cauchy–Schwarz
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In plain words

Engel-form Cauchy–Schwarz combines the three denominators into 2(x+y+z)2(x+y+z).

E≥x+y+z2E\ge\frac{x+y+z}{2}
Detailed analysis

Cauchy–Schwarz gives (y+z+z+x+x+y)E≥(x+y+z)2(y+z+z+x+x+y)E\ge(x+y+z)^2. Since the parenthesis is 2(x+y+z)2(x+y+z), we obtain E≥x+y+z2E\ge\frac{x+y+z}{2}.