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

Abstract

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, ∞). Forα ∈ [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.