 Therefore, the map $$H_{\gamma_0}(\alpha):=h_{\gamma_0}(\alpha)-\alpha\quad\text{with}\quad h_{\gamma_0}(\alpha):=\ophi^{\alpha,\beta(\alpha)}(\sigma\ud{u})$$ is continuous and strictly decreasing on $[\alpha_0,\gamma_0)$, $H_{\gamma_0}(\alpha_0)>0$ and $\lim_{\alpha\uparrow\gamma_0}H_{\gamma_0}(\alpha)