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