On some inequalities for the two-parameter Mittag-Leffler function in the complex plane

Starting from the well-known relationship $|{\mathrm{e}}^z| = {\mathrm{e}}^{{\mathrm Re}(z)}$, we consider the question whether $|E_{\alpha,\beta}(z)|$ and $E_{\alpha,\beta}({\mathrm Re}(z))$ are comparable, as functions of the complex variable $z$, where $E_{\alpha,\beta}$ denotes the two-parameter Mittag-Leffler function, a generalization of the exponential function. For some ranges of the parameters $\alpha$ and $\beta$ we prove inequalities between $|E_{\alpha,\beta}(z)|$ and $E_{\alpha,\beta}({\mathrm Re}(z))$ holding globally for all $z\in \mathbb{C}$. In some other ranges of $\alpha$ and $\beta$ the same inequalities are proved to hold asymptotically, i.e. for sufficiently small or large $z$. There are moreover some values of $\alpha$ and $\beta$ for which the situation is less clear, and some conjectures, motivated by numerical observations, are proposed.