\[\left|\varepsilon_{n}(z)\right|,~\beta^{-1}\left|\varepsilon_{n}^{\prime}(z)%
\right|\leq 2\alpha C_{n}\left|\frac{{\left(a\right)_{n}}{\left(a-b+1\right)_{%
n}}}{n!z^{a+n}}\right|\exp\left(\frac{2\alpha\rho C_{1}}{|z|}\right),\]