Let $f = x_1^2 + x_2^4$. How to find class $\mathcal{K_{\infty}}$ functions $\alpha_1(||x||)$ and $\alpha_2(||x||)$, such that,
$\alpha_1(||x||) \leq f \leq \alpha_2(||x||), \forall x \in {R}^2.$
$\textbf{Edit:}$
I have been following the nonlinear system analysis book by Khalil. According to the proof (Appendix C.4). One first needs to find,
$\phi(s) = \underset{s\leq ||x|| \leq r}{\text{inf}} f(x) \text{ for } 0\leq s\leq r.$
Then one can define $\alpha_1(s) \leq k\phi(s)$ with $0<k<1$.
But I am not able to understand how to find such $\phi(s)$ for given $f(s)$? I think there should a methodology to go about these types of questions. I would be grateful if someone helps me understand this.