I refer to the paper Goldfeld, D.; Sarnak, P. Sums of Kloosterman sums. Invent. Math. 71 (1983), no. 2, 243--250. They study the usual Kloosterman zeta function $Z_{m,n}(s,\chi)$, whose precise definition is not important here. Using the estimate $$ |Z_{m,n}(s,\chi)|= O\Big(\frac{|s|^{1/2}}{\sigma-1/2}\Big), \qquad s=\sigma + it, \sigma>\frac12, $$ they claim that as an application of the Phragmen-Lindelöf, for any $\epsilon>0$, we have $$ \Big|Z_{m,n}\Big(\frac{1+s}{2},\chi\Big)\Big| \ll |t|^{\frac12-\frac{\sigma}{2\beta}+\epsilon},\qquad 0<\epsilon \le \sigma\le \beta +\epsilon. $$ The Phragmen-Lindelöf principle is the following: Let $f(s)$ be a regular function of finite order in $\beta_1\le \sigma\le\beta_2$, and $f(s) = O(|t|^{k_i})$ for $\sigma=\beta_i$, with $i=1,2$. Then $$ f(s) = O(|t|^{k(\sigma)}) $$ uniformly for $\beta_1\le \sigma\le\beta_2$, where $k(x)$ is the linear function such that $k(\beta_1) = k_1$ and $k(\beta_2)=k_2$.
Question: How do I express the first big-$O$ estimate in the shape $O(|t|^{k})$? Presumably it is because of this that I am not able to see what how function $k(\sigma)$ in this case should be derived.