MathLabs

Problem 2

Let a,b,c be positive real numbers with abc=8. Prove that a squared divided by the square root of (1+a cubed)(1+b cubed), plus the two cyclic analogues, is at least 4/3.
Step 2 of 5: Apply the lemma cyclically
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a2(1+a3)(1+b3)≥2a2(2+a2)(2+b2)\dfrac{a^2}{\sqrt{(1+a^3)(1+b^3)}}\ge\dfrac{2a^2}{(2+a^2)(2+b^2)}
Detailed analysis

Apply the lemma to a and b in the first summand, and cyclically to the other two. Thus the original left side is at least the sum of 2a squared over (2+a squared)(2+b squared) and its cyclic analogues.