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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.

Text Solution

Verified by Experts

Let y =`f(x)=x^(m),where x gt 0 and 0 lt mlt1`
`(d^(2)y)/(dx^(2))=m(m-1)x^(m-2lt0`
So graph is concave downward
Therefore `(a_(1)^(m)+a_(2)^(m)+…a_(n)^(m))/(n)lt(a_(1)+a_(2)+…a_(n))/(n)^(n)`
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