<s>[INST] A tricky inequality: $n(n+1)^{\frac{1}{n}}+(n-2)n^{\frac{1}{n-2}}>2n,\ n\geq3.$ Let $H_n=1+\frac{1}{2}+\cdots+\frac{1}{n}$ be the $n-$th harmonic number,
it is not difficult to prove that:
(1) $n(n+1)^{\frac{1}{n}}<n+H_n,$ for $n>1$; (use AM-GM inequality)
(2) $(n-2)n^{\frac{1}{n-2}}>n-H_n$, for $n>2$. (Hint: $n=3$ is obvious,
when $n>3$, $n-(n-2)n^{\frac{1}{n-2}}<2<H_n$)
So from (1), we know
$$H_n>n(n+1)^{\frac{1}{n}}-n, n>1;\ (3)$$
from (2), we know
$$H_n>n-(n-2)n^{\frac{1}{n-2}},n>2.\ (4)$$
It seems inequality $(3)$ is better than $(4)$, that is to say,
$$\boxed{n(n+1)^{\frac{1}{n}}+(n-2)n^{\frac{1}{n-2}}>2n,\ n\geq3.}$$
My concern is whether there are tricky proofs for above inequality without complicated derivative calculations.
Any helps, hints and comments will welcome.
 [/INST] By Bernoulli
$$n(n+1)^{\frac{1}{n}}+(n-2)n^{\frac{1}{n-2}}=\frac{n}{\left(1+\frac{1}{n+1}-1\right)^{\frac{1}{n}}}+\frac{n-2}{\left(1+\frac{1}{n}-1\right)^{\frac{1}{n-2}}}\geq$$
$$\geq\frac{n}{1-\frac{1}{n+1}}+\frac{n-2}{1+\frac{1}{n-2}\cdot\frac{1-n}{n}}=n+1+\frac{n(n-2)^2}{n^2-3n+1}=$$
$$=2n+\frac{1}{n^2-3n+1}>2n$$
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