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Prove that (a(1)^(m)+a(2)^(m)+….+a(n)^(m...

Prove that `(a_(1)^(m)+a_(2)^(m)+….+a_(n)^(m))/(n)lt(a_(1)+a_(2)+..+a_(n))/(n)^(m)`
If `0ltmlt1 and a_(i)gt0` for all I.

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Consider function y=f(x) =`x^(m)` where x `gt` 0 and m`lt` 0 or m`gt`1
`therefore (dy)/(dx)=mx^(m-1)` and `(d^(2)y)/(dx^(2))=m(m-1)x^(m-2)`
since x `gt` 0 and x`lt`0 or m `gt` 1,`(d^(2)y)/(dbx^(2))gt0`
So, graph of function is concave upward.
Therefore `(overset(n)underset(i=1)Sigmaf(x_(1)))/(n)gtf((overset(n)underset(n)Sigma x_(1))/(n))`
or `a_(1)^(m)+a_+(2)^(m)+...A_(n)^(m)/(n)gt((a_(1)+a_(2)+...a_(n))/(n^(m)))`
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