Can Minkowski's Inequality Prove Summation Inequality for Positive Numbers?

[St41n]

I don't understand how it is possible to show using the Minkowski's Inequality that
$(\sum x_i )^a \leq \sum x_i^a$ where $x_i \geq 0 \forall i$ and $0<a<1$.

I also tried to prove this without using Minkowski, but to no avail.

This is driving me crazy although it seems to be trivial in the literature.
I will appreciate any help

 

[micromass]

Hi St41n!

It seems that you must prove that $(x+y)^\alpha\leq x^\alpha+y^\alpha$ for $x,y\geq 0$ and $0<\alpha<1$.

For that, you must look at the function

$$f:\mathbb{R}^+\rightarrow \mathbb{R}:x\rightarrow 1+x^\alpha-(1+x)^\alpha$$

Try to show that f is increasing and has its minimum in 0. It follows that $f(x)\geq 0$.

 

[St41n]

Thank you very much for the quick reply!