Abstract
One considers the class S of functions, regular and univalent in ¦Z¦<1 and normalized by the expansion f(z)=Z + C2Z2 +.... By the logarithmic coefficients of the function f (z)ɛ S one means the coefficients of the expansion
Earlier, the author had formulated the following conjecture: for any function f(z)ɛ S, for each z ɛ (0,1) one has the inequality
In this paper this conjecture is proved for spiral-shaped functions and for functions from S with real coefficients and under some additional assumptions.
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Literature cited
I. M. Milin,“On a certain property of the logarithmic coefficients of univalent functions,” in: Metric Questions of the Theory of Functions [in Russian], Naukova Dumka, Kiev (1980), pp. 86–90.
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A. Z. Grinshpan,“An application of the arc principle to Bieberbach-Eilenberg functions,” Mat. Zametki,11, No. 6, 609–618 (1972).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 135–143, 1983.
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Milin, I.M. A conjecture regarding the logarithmic coefficients of univalent functions. J Math Sci 26, 2391–2397 (1984). https://doi.org/10.1007/BF01680020
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DOI: https://doi.org/10.1007/BF01680020