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

TL;DR

This paper investigates inequalities involving the two-parameter Mittag-Leffler function in the complex plane, establishing conditions for their validity based on monotonicity and zero distribution, with implications for its convexity properties.

Contribution

It provides new necessary and sufficient conditions for inequalities involving the Mittag-Leffler function, linking them to monotonicity and zero-free regions based on parameters.

Findings

The inequality |E_{α,β}(z)| ≤ E_{α,β}(Re z) holds iff E_{α,β}(-x) is completely monotone.

Complete monotonicity of 1/E_{α,β}(x) is necessary for certain inequalities when α∈[1,2).

Absence of non-real zeros of E_{α,β} ensures inequalities for α≥2.

Abstract

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.