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 3 of 5: Bound the pair involving b and c
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(b+c/2)n>bn+cn  ⟹  bn+cnn<b+c/2(b+c/2)^n>b^n+c^n\implies\sqrt[n]{b^n+c^n}<b+c/2
Detailed analysis

Since b≥c>0b\ge c>0 and n≥2n\ge2, write t=c/b∈(0,1]t=c/b\in(0,1]. The binomial expansion gives (1+t/2)n>1+tn(1+t/2)^n>1+t^n, because its first extra term is nt/2≥tnnt/2\ge t^n and the remaining terms are positive. Multiplying by bnb^n yields (b+c/2)n>bn+cn(b+c/2)^n>b^n+c^n, so taking nnth roots gives bn+cnn<b+c/2\sqrt[n]{b^n+c^n}<b+c/2.