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

For the two-parameter Mittag-Leffler function $E_{\alpha ,\beta }$ with $\alpha >0$ and $\beta \ge 0$, we consider the question whether $\left|E_{\alpha ,\beta }\right(z\left)\right|$ and $E_{\alpha ,\beta }(\Re z)$ are comparable on the whole complex plane. We show that the inequality $\left|E_{\alpha ,\beta }\right(z\left)\right|\le E_{\alpha ,\beta }(\Re z)$ holds globally if and only if $E_{\alpha ,\beta }(-x)$ is completely monotone on $(0,\infty )$. For $\alpha \in [1,2)$ we prove that the complete monotonicity of $1/E_{\alpha ,\beta }\left(x\right)$ on $(0,\infty )$ is necessary for the global inequality $\left|E_{\alpha ,\beta }\right(z\left)\right|\ge E_{\alpha ,\beta }(\Re z)$, and also sufficient for $\alpha =1$. For $\alpha \ge 2$ we show that the absence of non-real zeros for $E_{\alpha ,\beta }$ is sufficient for the global inequality $\left|E_{\alpha ,\beta }\right(z\left)\right|\ge E_{\alpha ,\beta }(\Re z)$, and also necessary for $\alpha =2$. All these results have an explicit description in terms of the values of the parameters $\alpha ,\beta$. Along the way, several inequalities for $E_{\alpha ,\beta }$ on the half-plane $\{\Re z\ge 0\}$ are established, and a characterization of its log-convexity and log-concavity on the positive half-line is obtained.