Abstract
We give a short proof that the quartic polynomial \(f(z)=\frac 1 6 z^{4} + \frac 2 3 z^{3} + \frac 7 6 z^{2} + z\) is univalent, i.e., injective, in the open unit disc \(D=\{ z \in \mathbb {C} : \lvert z \rvert <1 \}\).
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References
J.W. Alexander, Functions which map the interior of the unit circle upon simple regions. Ann. Math. 17, 12–22 (1915)
D. Dmitrishin et al., Estimating the Koebe radius for polynomials (2018). ArXiv:1805.06927
A. Gluchoff, F. Hartmann, Univalent polynomials and non-negative trigonometric sums. Am. Math. Mon. 105, 508–522 (1998)
A. Gluchoff, F. Hartmann, On a “Much Underestimated” paper of Alexander. Arch. Hist. Exact Sci. 55(1), 1–41 (2000)
A. Gluchoff, F. Hartmann, Zero sets of polynomials univalent in the unit disc. e-script (2002)
A. Gluchoff, F. Hartmann, A “forceful” construction of 1-1 complex polynomial mappings. e-script (2006)
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Dillies, J. (2019). Univalence of a Certain Quartic Function. In: Abell, M., Iacob, E., Stokolos, A., Taylor, S., Tikhonov, S., Zhu, J. (eds) Topics in Classical and Modern Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-12277-5_6
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DOI: https://doi.org/10.1007/978-3-030-12277-5_6
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