I want to know under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where

${E_{\alpha ,1}}(z) = \sum\limits_{k = 0}^\infty {\frac{{{z^k}}}{{\Gamma (\alpha k + 1)}}}$.

${E_{1,1}}(z) = \sum\limits_{k = 0}^\infty {\frac{{{z^k}}}{{\Gamma (k + 1)}}} {\text{ = }}{{\text{e}}^z}$ or ${E_{0,1}}(z) = \frac{1}{{1 - z}}$ has no zero, how about $\alpha \in (0,1)$?