Positivity and asymptotics for the smallest positive zero of E_{α,α}(-z^α)

Establish the conjectured properties of the two-parameter Mittag-Leffler function E_{α,α} evaluated at −z^α for α in (1,2]: specifically, prove that E_{α,α}(−z^α) > 0 for all z in [0, 2.93785]; and determine the asymptotic behavior of Z_{α}(α), defined as the smallest positive real number z such that E_{α,α}(−z^α) = 0, namely Z_{α}(α) = (c ln(α − 1) + d)(1 + o(1)) as α → 1+ with constants c ≈ −0.81 and d ≈ 2.25, and Z_{α}(α) = π + c(2 − α) + o(2 − α) as α → 2− with c ≈ −0.81.

Background

Section 3 motivates the need for quantitative knowledge of small positive zeros of E_{α,β}(−z^α) because the separation results (Section 2) rely on choosing T* so that certain Mittag-Leffler kernels remain positive. For β = α, the authors computed numerical evidence for the smallest positive zero Z_{α}(α) and observed non-monotonic dependence on α.

Based on these observations, they formulate a conjecture combining a uniform positivity claim on [0, 2.93785] for all α ∈ (1,2] and two asymptotic regimes of Z_{α}(α) near α → 1+ and α → 2−, with approximate constants inferred from computations. Proving these assertions would directly strengthen the applicability of their separation theorems by certifying the positivity intervals analytically.