MathLabs

Problem 4

Let a,b,ca,b,c be the side lengths of a triangle with a+b+c=1a+b+c=1, and let n≥2n\ge2 be an integer. Prove that an+bnn+bn+cnn+cn+ann<1+21/n2\sqrt[n]{a^n+b^n}+\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<1+\frac{2^{1/n}}2.
Step 4 of 5: Bound the pair involving c and a
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(c+a/2)n>cn+an  ⟹  cn+ann<a+c/2(c+a/2)^n>c^n+a^n\implies\sqrt[n]{c^n+a^n}<a+c/2
Detailed analysis

Apply the same estimate with the pair (a,c)(a,c): since a≥c>0a\ge c>0, (c+a/2)n>cn+an(c+a/2)^n>c^n+a^n, and therefore cn+ann<a+c/2\sqrt[n]{c^n+a^n}<a+c/2.