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.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 7, pp. 987–994, July, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i7.7222.
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Hu, Z., Fan, J. & Wang, X. Univalence Criteria for Locally Univalent Analytic Functions. Ukr Math J 75, 1128–1137 (2023). https://doi.org/10.1007/s11253-023-02250-2
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DOI: https://doi.org/10.1007/s11253-023-02250-2