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 2 of 5: Bound the pair of largest sides
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an+bnn<2ann=21/na<21/n2\sqrt[n]{a^n+b^n}<\sqrt[n]{2a^n}=2^{1/n}a<\frac{2^{1/n}}2
Detailed analysis

Because a,b>0a,b>0 and b≤a<1/2b\le a<1/2, we have an+bn≤2ana^n+b^n\le2a^n and therefore an+bnn≤21/na<21/n/2\sqrt[n]{a^n+b^n}\le2^{1/n}a<2^{1/n}/2.