\left\vert {\operatorname{Im}}\lambda \right\vert \cdot x_{n}-\widetilde{c}r^{-\varepsilon
}x_{n}^{\varepsilon }& =\left( \left\vert {\operatorname{Im}}\lambda \right\vert \cdot
x_{n}-\delta \widetilde{c}r^{-\varepsilon }x_{n}^{\varepsilon }\right)
-(1-\delta )r^{-\varepsilon }x_{n}^{\varepsilon }  \label{eq6'''.3} \\
& =\left\vert {\operatorname{Im}}\lambda \right\vert x_{n}^{\varepsilon }\left(
x_{n}^{1-\varepsilon }-\delta \frac{\widetilde{c}}{\left\vert {\operatorname{Im}}\lambda
\right\vert }\right) -(1-\delta )r^{-\varepsilon }x_{n}^{\varepsilon }. 
\notag\end{aligned}$$By the right hand side of doesn’t exceed $$\begin{aligned}
&&\left\vert {\operatorname{Im}}\lambda \right\vert x_{n}^{\varepsilon }\left( \frac{\widetilde{c}}{\left\vert {\operatorname{Im}}\lambda \right\vert }\frac{1}{r^{\varepsilon
+1}}-\frac{\widetilde{c}}{\left\vert {\operatorname{Im}}\lambda \right\vert }\frac{\delta }{r^{\varepsilon }}\right) -(1-\delta )r^{-\varepsilon }x_{n}^{\varepsilon } \\
&=&-\widetilde{c}\left( \delta -\frac{1}{r}\right) x_{n-1}^{\varepsilon
}-(1-\delta )x_{n-1}^{\varepsilon } \\
&<&-\limfunc{const}x_{n-1}^{\varepsilon }.\end{aligned}$$It follows now from this estimate and that $$\left\vert \widetilde{G}(\lambda )\right\vert \lesssim _{g}\sum_{n\geq
0}\exp \{-\limfunc{const}x_{n}^{\varepsilon }\}<\infty .  \label{eq6iv.0}$$Similarly one proves that all derivatives of $G$ are also bounded on $\mathbb{C}^{+}$.

It remains now to show . One has $$\begin{aligned}
\left\vert \partial _{\overline{\lambda }}\widetilde{G}\right\vert & \leq
\sum_{n\geq 1}\theta ^{\prime }\left( r^{\varepsilon +2}x_{n}^{1-\varepsilon
}\frac{\left\vert {\operatorname{Im}}\lambda \right\vert }{\widetilde{c}}\right) \frac{r^{\varepsilon +1}x_{n}^{1-\varepsilon }}{2\widetilde{c}}\left\vert
G_{n}\right\vert  \label{eq6iv.1} \\
& \lesssim _{g}\sum_{n\geq 1}x_{n}^{1-\varepsilon }\exp \{\left\vert {\operatorname{Im}}\lambda \right\vert \cdot x_{n}-\widetilde{c}r^{-\varepsilon
}x_{n}^{\varepsilon }\}.  \notag\end{aligned}$$ Only terms subject to $$\frac{\widetilde{c}r^{-\varepsilon -2}}{\left\vert {\operatorname{Im}}\lambda \right\vert }\leq x_{n}^{1-\varepsilon }\leq \frac{\widetilde{c}r^{-\varepsilon -1}}{\left\vert {\operatorname{Im}}\lambda \right\vert }  \label{eq6iv.2}$$make a non trivial contribution to the series in . The inequality implies $$\begin{aligned}
x_{n}& \geq \left( \frac{\widetilde{c}r^{-\varepsilon -2}}{\left\vert {\operatorname{Im}}\lambda \right\vert }\right) ^{\frac{1}{1-\varepsilon }},  \notag \\
\intertext{or}
x_{n}^{\varepsilon }& \geq \left( \frac{\widetilde{c}r^{-\varepsilon -2}}{\left\vert {\operatorname{Im}}\lambda 