($i=1\sim5$) in the Higgs potential, and $\lambda_i$ are related to the Higgs boson masses through Eqs.(\[lambda1\])-(\[lambda5\]), these constraints can be translated into the bounds on the masses $m_{\tilde H_1^0}$, $m_{\tilde H_1^{0'}}, m_{H_3}^{}$ and $m_{H_5}^{}$ and on the mixing angles $\theta_H$ and $\alpha$.

We here show the numerical results on the Higgs mass bounds. Fig. 1 shows the allowed regions of the masses in the $m_{H_3}^{}$ - $m_{H_5}^{}$ plane (a), in the $m_{H_3}^{}$ - $m_{\tilde H_1^0}$ plane (b) and in the $m_{\tilde H_1^0}$ - $m_{\tilde H_1^{0'}}$ plane (c). We vary the Higgs boson masses in the range $m_{H_3}^{}, m_{H_5}^{}, m_{\tilde H_1^0}, m_{\tilde H_1^{0'}}  <$ 1 TeV and the mixing angles for $0 < \theta_H \le\pi/2$ and $-\pi/2 < \alpha \le \pi/2$. In each figure, the conditions of perturbative unitarity in Eq.(\[PU\]) are satisfied inside the regions. In Fig. 1(a), light shadowed region is excluded by the $Z\to b\bar b$ result.

![Allowed regions of the masses of the Higgs bosons in the $m_{H_3}^{}$ - $m_{H_5}^{}$ plane (a), in the $m_{H_3}^{}$ - $m_{\tilde H_1^0}$ plane (b) and in the $m_{\tilde H_1^0}$ - $m_{\tilde H_1^{0'}}$ plane (c). In Fig. 1(a) the light shadowed regions are excluded by the $Zb\bar b$ results. []{data-label="figure"}](fig1a.eps "fig:"){width="4.9cm"}     ![Allowed 