Univalence Criteria for Locally Univalent Analytic Functions

Suppose that p(z) = 1 + zϕ″(z)/ϕ′(z), where ϕ(z) is a locally univalent analytic function in the unit disk D with ϕ(0) = ϕ′(1) − 1 = 0. We establish the lower and upper bounds for the best constants σ₀ and σ₁ such that \({e}^{{-\sigma }_{0}/2}<\left|p\left(z\right)\right|<{e}^{{\sigma }_{0}/2}\) and |p(w)/p(z)| < \({e}^{{\sigma }_{1}}\) for z, w ∈ D, respectively, imply the univalence of ϕ(z) in D.