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 $|E_{\alpha,\beta}(z)|$ and $E_{\alpha,\beta}(\Re z)$ are comparable on the whole complex plane. We show that the inequality $|E_{\alpha,\beta}(z)|\le E_{\alpha,\beta}(\Re z)$ 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 $1/E_{\alpha,\beta}(x)$ on $(0,\infty)$ is necessary for the global inequality $|E_{\alpha,\beta}(z)|\ge E_{\alpha,\beta}(\Re z),$ 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 $|E_{\alpha,\beta}(z)|\ge E_{\alpha,\beta}(\Re z),$ 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.