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

TL;DR

This work investigates when the Mittag-Leffler function $E_{\alpha,\beta}$ satisfies global comparisons between $|E_{\alpha,\beta}(z)|$ and $E_{\alpha,\beta}(\Re z)$ in the complex plane. It connects these global inequalities to complete monotonicity and Bernstein-function structure, employing probabilistic representations, Hadamard factorizations of zeros, and precise asymptotics to delineate regions in $(\alpha,\beta)$ where the inequalities hold, including $0<\alpha\le 1$, $\beta\ge\alpha$ for the forward inequality and special lines $\alpha=1$ and $\alpha=2$ for the reverse inequality. Key contributions include a complete CM criterion for $E_{\alpha,\beta}(-x)$ and $1/E_{\alpha,\beta}(x)$ in various parameter regimes, explicit log-convexity/log-concavity characterizations on $\mathbb{R}^+$, and open problems about global validity in the presence of non-real zeros. These results advance understanding of fractional-calculus kernels and provide a framework for analyzing stability and perturbation sensitivity of matrix Mittag-Leffler functions.

Abstract

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.

Figures (2)

• Figure 1: Regions in the parameter space $(\alpha,\beta)$ where $\tilde{E}_{\alpha,\beta}(x)=\Gamma(\beta) E_{\alpha,\beta}(x)$ is super-additive (orange) or sub-additive (green) on $\mathbb{R}^{+}.$ The curve represents the separating function $\beta=h(\alpha)$. For $(\alpha,\beta)$ in the gray region, the function $\Gamma(\beta) E_{\alpha,\beta}(x)$ is neither super-additive nor sub-additive on $\mathbb{R}^{+}.$
• Figure 2: Regions (orange and green) in the parameter space $(\alpha,\beta)$ where inequality \ref{['eq:Ineq_LE']} or \ref{['eq:Ineq_GE']}, respectively, holds globally, and regions (gray) where neither holds globally. The area with diagonal gray narrow lines corresponds to the region where we conjecture that no inequality holds globally, whereas areas with green spaced diagonal lines correspond to regions where we conjecture that inequality \ref{['eq:Ineq_GE']} holds globally. The blue curve represents the function $\beta=h(\alpha)$ from Lemma \ref{['lem:h_Function']}, while the dashed green curve represents a possible example of the unknown increasing convex function from Conjecture \ref{['conj:CBF']}.

Theorems & Definitions (39)

• Theorem 1
• proof
• Remark 2
• Proposition 3
• proof
• Remark 4
• Corollary 5
• proof
• Remark 6
• Corollary 7