fraction power vector-norm inequality If X is a Banach Space and $x,y\in X$. Then by the definition of a Banach algebra we know $$\|x.y\|\leq\|x\|\|y\|$$ 
and thats how we can have relation for any positive power. i.e. $n\in N$, 
$$\|x^n\|\leq \|x\|^n.$$
Now the question is, what would be the relation for a fraction power? say, $\frac{1}{n}$??  and what would be if power is some real number? not an integer. 
I tried it and got
$$\|x^{1/r}\|\geq \|x\|^{1/r},\quad r\in Z^+$$ and $$\|x^{1/r}\|\leq \|x\|^{1/r},\quad r\in Z^-$$ but I am not sure if it is so. 
it may seems a trivial one but its a troublemaker for me. 

If, for $n \in \mathbb Z^+$, the notation $x^{1/n}$ makes sense, then we have $(x^{1/n})^n = x$ and, hence,
$$\|x\| = \|(x^{1/n})^n\| \le \|x^{1/n}\|^n.$$
This proves
$$\|x\|^{1/n} \le \|x^{1/n}\|.$$
