Considering,

1. the set of all n dim. vectors $\{x_i\}_{i=1,...,n} $ such that $x_i \geq 0 $ and $\sum_{i=1}^{n}x_i = K$
2. Any continuous and strictly increasing function $f^+(x)$ : $ \mathbb R^+ \to \mathbb R^+ $ (i.e. where $f^+(x+\epsilon)> f^+(x) \quad \forall \epsilon >0 \quad x \in \mathbb R^+$)

My question: Is it true in general that $ \sum_{i=1}^n x_if(x_i)\geq Kf(\frac{K}{n})$ for any $\{x_i\}$ respecting 1. and $f^+$ ?

Using cauchy-schwartz ineq. I was able to prove this result for linear functions $f^+(x)=ax+b, \quad a,b>0$ , then:

$\sum_{i=1}^n x_if(x_i)=Kb+a\sum_{i=1}^n x_i^2\geq Kb+\frac{aK^2}{n}$ since $n\sum_{i=1}^n x_i^2\geq (\sum_{i=1}^n x_i)^2=K^2$ .

(where $n\sum_{i=1}^n x_i^2=K^2$ if $x_i=K/n$)

Intuitively I was thinking that this should lead to a more general proof of the statement since for any interval $I=[x_{min},x_{max}]$, we can always find $a,b>0$ such that $ax+b\leq f^+(x) \quad \forall x \in I$. But however I didn't find yet the way to conclude using this path.

I am missing something? Is there another way to proceed?

Thank's for your help!

Considering,

1. the set of all n dim. vectors $\{x_i\}_{i=1,...,n} $ such that $x_i \geq 0 $ and $\sum_{i=1}^{n}x_i = K$
2. Any continuous and strictly increasing function $f^+(x)$ : $ \mathbb R^+ \to \mathbb R^+ $ (i.e. where $f^+(x+\epsilon)> f^+(x) \quad \forall \epsilon >0 \quad x \in \mathbb R^+$)

My question: Is it true in general that $ \sum_{i=1}^n x_if(x_i)\geq Kf(\frac{K}{n})$ for any $\{x_i\}$ respecting 1. and $f^+$ ?

Using cauchy-schwarz ineq. I was able to prove this result for linear functions $f^+(x)=ax+b, \quad a,b>0$ , then:

$\sum_{i=1}^n x_if(x_i)=Kb+a\sum_{i=1}^n x_i^2\geq Kb+\frac{aK^2}{n}$ since $n\sum_{i=1}^n x_i^2\geq (\sum_{i=1}^n x_i)^2=K^2$ .

(where $n\sum_{i=1}^n x_i^2=K^2$ if $x_i=K/n$)

Intuitively I was thinking that this should lead to a more general proof of the statement since for any interval $I=[x_{min},x_{max}]$, we can always find $a,b>0$ such that $ax+b\leq f^+(x) \quad \forall x \in I$. But however I didn't find yet the way to conclude using this path.

I am missing something? Is there another way to proceed?

Thank's for your help!