Let $\phi:\mathbb R \to \mathbb R$ smooth function satisfying
(1) there exists $m_1$ such that $|\phi^{(\mu)}(r)| \leq C r^{m_1-\mu}$ for all $\mu \in \mathbb N\cup \{ 0\}$ for $r\geq 1$
(2) there exists $m_1$ such that $|\phi^{(\mu)}(r)| \leq C_1 r^{m_2-\mu}$ for all $\mu \in \mathbb N\cup \{ 0\}$, $0<r<1$.
Let $h\in C^{\infty}(\mathbb R^d\setminus \{0\})$ is positive homogeneous function of degree $\lambda>0$ such that $m_1\lambda\leq 2.$
Consider composition fucntion $g=\phi\circ h:\mathbb R^d \to \mathbb R.$
I'm looking for the examples of $g$ of the above form.
For example: Let $h(\xi)=|\xi|$ and $\phi(r)=r^{\alpha} (\alpha>0).$ Then $g(\xi)=|\xi|^{\alpha} \ (\alpha>0)$
My questions: (1) For which $\alpha>0,$ $g(\xi)$ satisfies the above conditions? (2) Can we construct few more concrete examples? (3) Is real polynomial of degree $\lambda$ satisfies the above conditions?