Koebe domains for the class of typically real odd functions

In this paper we discuss the generalized Koebe domains for the class $T ^{(2)}$ and the set $D\subset \Delta=\{z\in \mathbb{C}:|z|< 1\}$, i.e. the sets of the form $\cap_{f\in TM} f(D)$. The main idea we work with is the method of the envelope. We determine the Koebe domains for $H=\{z\in \Delta : |z^{2}+1|>2|z|\}$ and for special sets $\Omega_{\alpha}, \alpha \le \frac{4}{3}$. It appears that the set $\Omega_{\frac{4}{3}}$ is the largest subset of $\Delta$ for which one can compute the Koebe domain with the use of this method. It means that the set $K_{T^{(2)}}(\Omega_{\frac{4}{3}})\cup K_T (\Delta)$ is the largest subset of the still unknown set $K_{T^{(2)}}(\Delta)$ which we are able to derive.