Prove that $$ \left( \frac{a+b+c\cdots k}{n} \right) ^{a+b+c \cdots k} \lt a^a b^b c^c \cdots k^k $$
We can prove that $$ (1+x)^{1+x} (1-x)^{1-x} \gt 1$$ (if $x \lt 1$) and from there by assuming $x = \frac{u}{z}$ and then $u+z =a$ and $u-z =b$ we will have $$ \left( \frac{a+b}{2}\right)^{a+b} \lt a^a b^b$$ but how to extend this result upto $n$ number of terms?
