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

For the two-parameter Mittag-Leffler function Eα,β with α>0 and β≥0, we consider the question whether |Eα,β(z)| and Eα,β(ℜz) are comparable on the whole complex plane. We show that the inequality |Eα,β(z)|≤Eα,β(ℜz) holds globally if and only if Eα,β(−x) is completely monotone on (0,∞). For α∈[1,2) we prove that the complete monotonicity of 1/Eα,β(x) on (0,∞) is necessary for the global inequality |Eα,β(z)|≥Eα,β(ℜz), and also sufficient for α=1. For α≥2 we show that the absence of non-real zeros for Eα,β is sufficient for the global inequality |Eα,β(z)|≥Eα,β(ℜz), 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 Eα,β on the half-plane {ℜz≥0} are established, and a characterization of its log-convexity and log-concavity on the positive half-line is obtained.