On Bounds for the Complex Lambert W − 1 W −1 Branch Near the Branch Cut

We study the modulus |W₋₁(z)| of the W₋₁ branch of the Lambert W function for complex arguments z in a punctured neighbourhood of the branch point z = −1/e. Writing z = −1/e + r·eiθ with 0 < r ≪ 1 and θ ∈ [0,2π), we derive the asymptotic expansion |W₋₁(z)| = 1 − √(2er) cos(θ/2) + O(r) and obtain the two‑sided bound 1 − √(2er) + O(r) ≤ |W₋₁(z)| ≤ 1 + √(2er) + O(r) in the asymptotic sense as r → 0⁺. The extremal values occur on opposite sides of the branch cut, at θ = 0 and θ → 2π⁻. Numerical verification is provided, and a conjecture on rigorous inequalities with an explicit constant is formulated. This work gives the first systematic description of |W₋₁| in the complex domain near the branch point, complementing classical real‑line results.