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                \begin{document}$$\begin{aligned}&|Z_{\rho ,\omega ,h} - 1|\leqslant C|U|,\quad |v_{\omega , h} - \frac{3}{2}|\leqslant C|U|,&\forall h^*_2\leqslant h\leqslant 0\;,\nonumber \\&|\xi _{-, h}|\leqslant C|U|2^{\theta h},\quad |\delta _{-, h}|\leqslant C|U|2^{\theta h},&\forall h^*_2\leqslant h\leqslant 0\;, \nonumber \\&|\xi _{+, h}-\xi _{-,h}|\leqslant C|U|2^{h^*_1-h},\quad |\delta _{+, h}-\delta _{-,h}|\leqslant C|U|2^{h^*_1-h},&\forall h^*_1\leqslant h\leqslant 0\;, \end{aligned}$$\end{document}$$for a suitable $\documentclass[12pt]{minimal}
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                \begin{document}$$C>0$$\end{document}$ (recall that, by definition, $\documentclass[12pt]{minimal}
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                \begin{document}$$\xi _{+,h}=\delta _{+,h}=Z_{\rho ,+,h}=v_{+,h}=0$$\end{document}$, $\documentclass[12pt]{minimal}
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                \begin{document}$$\forall h<h^*_1$$\end{document}$). Note that the bound on the beta function is already enough to prove the assumption for $\documentclass[12pt]{minimal}
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