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 $$\|x\|^{1/n} \le \|x^{1/n}\|.$$