On the real zeros of functions of Mittag-Leffler type

In the present paper, we prove an assertion allowing us to extend results related to the presence or absence of real zeros of functions of Mittag-Leffler type
$$ E_{1/\alpha}(z;\mu)=\sum_{k=0}^\infty\frac{z^k}{\Gamma(\alpha k+\mu)} $$
for certain values of $\alpha$ and $\mu$ to more extensive ranges of these parameters. We give a geometric description of the sets of pairs $(\alpha,\mu)$ for which the function $E_{1/\alpha}(z;\mu)$ has and does not have real zeros.