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Bazilevič theorem and the growth of univalent functions

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1275)

Abstract

One of the remarkable proofs of Milin is the elegant derivation of Hayman's regularity theorem for the coefficients of univalent functions. Milin's proof [4] does not contain any integration. The main tool is the theorem of Bazilevič [2].

In the following we prove a stronger version of an important theorem of Hayman on the growth of univalent functions.

Our proof is along Milin's approach. We also bring as an application a short proof of Hayman's regularity theorem [3] stating that

$$\left| {\tfrac{{^a n}}{{{\mathbf{ }}n}}} \right| \to \alpha = \alpha \left( f \right) for{\mathbf{ }}any{\mathbf{ }}f\varepsilon s{\mathbf{ }}as{\mathbf{ }}n \to \infty .$$

The research was partially supported by the Fund for the Promotion of Research at the Technion.

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References

  1. Aharonov, D., Bazilevič theorem for areally p-valent functions, J. London Math. Soc. (2), 27 (1983).

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  2. Bazilevič, I. E., On a univalence criterion for regular functions and the dispersion of their coefficient, Mat. Sb. 74 (1967), 135–146.

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  3. Hayman, W. K., Multivalent functions, 1st edition, Cambridge University Press, Cambridge, 1958.

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  4. Milin, I. M., Hayman's regularity theorem for the coefficients of univalent functions, Dokl. Adad. Nauk SSR, 192 (1970), 738–741; Soviet Math. Dokl. 11 (1970), 724–728.

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Dedicated to Maurice Heins

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© 1987 Springer-Verlag

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Aharonov, D. (1987). Bazilevič theorem and the growth of univalent functions. In: Berenstein, C.A. (eds) Complex Analysis I. Lecture Notes in Mathematics, vol 1275. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078340

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  • DOI: https://doi.org/10.1007/BFb0078340

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18356-3

  • Online ISBN: 978-3-540-47899-7

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