An induction proof of $(a+1)^n (a^n+1) \leq 2^n (a^{2n}+1)$

Question

Prove by induction that for every $n \in \mathbb{N} $ $$ (a+b)^n (a^n+b^n) \leq 2^n (a^{2n} +b^{2n}) \qquad a, b>0. $$ We can remark this equality is equivalent to $$ (a+1)^n (a^n+1) \leq 2^n (a^{2n}+1) \qquad a>0. $$

Answer 1

For $n=1$ we have $$2(a^2+1)\geq(a+1)^2$$ or $$(a-1)^2\geq0.$$ Let $$2^n\left(a^{2n}+1\right)\geq(a+1)^n\left(a^n+1\right).$$ Thus, it remains to prove that $$2^{n+1}\left(a^{2n+2}+1\right)\geq(a+1)^{n+1}\left(a^{n+1}+1\right)$$ and since $$(a+1)^n\leq\frac{2^n\left(a^{2n}+1\right)}{a^n+1},$$ it's enough to prove that $$2^{n+1}\left(a^{2n+2}+1\right)\geq\frac{2^n\left(a^{2n}+1\right)(a+1)\left(a^{n+1}+1\right)}{a^n+1}.$$ But $$a^{2n+2}+1\geq\frac{\left(a^{2n+1}+1\right)\left(a^{n+1}+1\right)}{a^n+1}$$ it's $$\left(a^{n+1}-1\right)(a-1)\geq0,$$ which is obvious.

Thus, it's enough to prove that $$2\left(a^{2n+1}+1\right)\geq\left(a^{2n}+1\right)(a+1)$$ or $$\left(a^{2n}-1\right)(a-1)\geq0.$$ Done!