1. ## Convexity of function

Hi all,

Can anybody show that the function $\displaystyle f: R^N \rightarrow R$ is convex, where:

$\displaystyle f(x) = \frac{K + \sum_{i=1}^N x_i^2 }{ \sum_{i=1}^N x_i }$

And $\displaystyle x_i \geq 0, \forall i$ and $\displaystyle K > 0$

Any help is welcome (this is not homework!).

Thanks!

2. Originally Posted by leoemil
Hi all,

Can anybody show that the function $\displaystyle f: R^N \rightarrow R$ is convex, where:

$\displaystyle f(x) = \frac{K + \sum_{i=1}^N x_i^2 }{ \sum_{i=1}^N x_i }$

And $\displaystyle x_i \geq 0, \forall i$ and $\displaystyle K > 0$

Any help is welcome (this is not homework!).

Thanks!
The not thinking way is to show that the Hessian $\displaystyle D_iD_jf(x)$ is positive semi-definite.