The second coefficient of univalent Bieberbach-Eilenberg functions near the identity

Hummel, J. A.

doi:10.1090/S0002-9939-1980-0577751-9

The second coefficient of univalent Bieberbach-Eilenberg functions near the identity
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Proc. Amer. Math. Soc. 80 (1980), 237-243 Request permission

Abstract:

An asymptotic expansion is found for the maximum $|{b_2}|$, in terms of $\beta = 1 - |{b_1}|$, for functions $F(z) = {b_1}z + {b_2}{z^2} + \cdots$ which are in the class of univalent Bieberbach-Eilenberg functions and which are near the identity (${b_1}$ near 1). The first two terms of this expansion are the same as the expansion of $|{b_2}|$ in terms of $1 - {b_1}$ for the functions which map the unit disc onto the interior of circles passing through $\pm 1$.

References

Ludwig Bieberbach, Über einige Extremalprobleme im Gebiete der konformen Abbildung, Math. Ann. 77 (1916), no. 2, 153–172 (German). MR 1511853, DOI 10.1007/BF01456900
Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773

Sur quelques propriétés topologiques de la surface de sphère

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P. R. Garabedian and M. Schiffer, The local maximum theorem for the coefficients of univalent functions, Arch. Rational Mech. Anal. 26 (1967), 1–32. MR 225994, DOI 10.1007/BF00283856
J. A. Hummel, The $b_{1},\,b_{2}$ coefficient body for Bieberbach-Eilenberg functions, J. Analyse Math. 33 (1978), 168–190. MR 516046, DOI 10.1007/BF02790172
J. A. Hummel and M. M. Schiffer, Variational methods for Bieberbach-Eilenberg functions and for pairs, Ann. Acad. Sci. Fenn. Ser. A I Math. 3 (1977), no. 1, 3–42. MR 0584969
James A. Jenkins, On Bieberbach-Eilenberg functions, Trans. Amer. Math. Soc. 76 (1954), 389–396. MR 62831, DOI 10.1090/S0002-9947-1954-0062831-6
James A. Jenkins, On Bieberbach-Eilenberg functions. III, Trans. Amer. Math. Soc. 119 (1965), 195–215. MR 181742, DOI 10.1090/S0002-9947-1965-0181742-6
Reiner Kühnau, Geometrie der konformen Abbildung auf der projektiven Ebene, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 12 (1963), 5–19 (German). MR 166342