Prove $ a^n+b^{n} \geq a^{k}b^{n-k} + a^{n-k}b^{k} $

Question

can someone please help me to prove this following inequality.

for $a,b > 0$ and $n,k > 1$

$ a^n+b^{n} \geq a^{k}b^{n-k} + a^{n-k}b^{k} $

Thanks in advance

Answer 1

For $n\geq k>0$ we obtain: $$a^n+b^n-a^kb^{n-k}-a^{n-k}b^k=(a^k-b^k)(a^{n-k}-b^{n-k})\geq0.$$ For $0<n<k$ it's wrong. Try, $n=1$ and $k=2$.