Proof of the Zalcman conjecture for initial coefficients
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Abstract The well-known Zalcman conjecture, which implies the Bieberbach conjecture, states that the coefficients of univalent functions on the unit disk satisfy the inequality for all n > 2, with the equality only for the Koebe function. This conjecture remained open for n > 3. We provide here the proof of this inequality for n = 4, 5, 6. It relies on the holomorphic homotopy of univalent functions and comparison of generated singular conformal metrics in the disk. The extremality of Koebe's function follows from hyperbolic properties.
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2019 ◽
Vol 12
(02)
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pp. 1950017
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1986 ◽
Vol 38
(6)
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pp. 1329-1337
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