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 5 of 5: Add the bounds
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bn+cnn+cn+ann<b+c/2+a+c/2=1\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<b+c/2+a+c/2=1
Detailed analysis

Adding the last two inequalities gives bn+cnn+cn+ann<a+b+c=1\sqrt[n]{b^n+c^n}+\sqrt[n]{c^n+a^n}<a+b+c=1. Combining this with the strict first bound yields the required inequality.