> 
> Let ${p\_{1}},{p\_{2}},\ldots,{p\_{n}}$ and
>  ${a\_{1}},{a\_{2}},\ldots,{a\_{n}}$ be positive real numbers and let
>  $r$ be a real number. Then for $r\ne0$ , we define
>  ${M\_{r}}(a,p)={\left({\frac{{{p\_{1}}a\_{1}^{r}+{p\_{2}}a\_{2}^{r}+\cdots+{p\_{n}}a\_{n}^{r}}}{{{p\_{1}}+{p\_{2}}+\cdots+{p\_{n}}}}}\right)^{1/r}}$
>  and for $r=0$ , we define
>  ${M\_{0}}(a,p)={\left({a\_{1}^{{p\_{1}}}a\_{2}^{{p\_{2}}}\cdots
>  a\_{n}^{{p\_{n}}}}\right)}^{1/\sum\nolimits \_{i=1}^{n}p\_i}$ . Then prove that $
>  {M\_{{k\_{1}}}}(a,p)\geqslant{M\_{{k\_{2}}}}(a,p) $ if $k\_{1}\geqslant
>  k\_{2}$.
> 
> 
> 

How to prove this generalized theorem? I have found this in a book without any proof. So can anyone show me?