A coefficient problem with typically real extremal function
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- by Seiji Konakazawa PDF
- Proc. Amer. Math. Soc. 123 (1995), 2723-2730 Request permission
Abstract:
Let $\Sigma _{0}$ denote the class of univalent functions in $|z| > 1$, with expansion $f(z) = z + \sum \nolimits _{n = 1}^\infty {{b_n}{z^{ - n}}}$. We show that if the omitted set of an $f(z) \in {\sum _0}$ is on the trajectory arcs of the quadratic differential $- w(w - \lambda )d{w^2}$ with $\lambda \geqq 4(\sqrt 2 - 1)$, then $f(z)$ has real coefficients. From this we can derive the coefficient estimate of ${\max _{{\Sigma _{0}}}} \mathcal {R}e( - {b_3} - \frac {1}{2}b_1^2 + \lambda {b_2})$.References
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- Seiji Konakazawa, Remarks on geometric properties of certain coefficient estimates, Kodai Math. J. 10 (1987), no. 2, 242–249. MR 897259, DOI 10.2996/kmj/1138037419 M. Schiffer, A method of variation within the family of simple functions, Proc. London Math. Soc. 44 (1938), 432-449.
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Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 2723-2730
- MSC: Primary 30C50; Secondary 30C70
- DOI: https://doi.org/10.1090/S0002-9939-1995-1257115-6
- MathSciNet review: 1257115