Geodesic
The generalized Koebe function
Problemy analiza, no. 17 (2010), pp. 61-66
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We observe that the extremal function for $|a_{3}|$ within the class $U'_{\alpha}$ (see Starkov [1]) has as well the property that max $|A_{4}|>4.15$, if $\alpha=2$. The problem is equivalent to the global estimate for Meixner-Pollaczek polynomials $P^{1}_{3}(x;\theta)$. 
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I. Naraniecka; J. Szynal; A. Tatarczak. The generalized Koebe function. Problemy analiza, no. 17 (2010), pp. 61-66. http://geodesic.mathdoc.fr/item/PA_2010_17_a5/

[1] Starkov V. V., “The estimates of coefficients in locally-univalent family $U'_{\alpha}$”, Vestnik Lenin. Gosud. Univ., 13 (1984), 48–54 (In Russian) | MR | Zbl

[2] Koekoek R., Swarttouw R. F., “The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue”, Report 98-17, Delft University of Technology, 1998

[3] Starkov V. V., Linear-invariant families of functions, Dissertation, Ekatirenburg, 1999, 287 pp. (In Russian)