Geodesic
The Radius of Convexity of a Linear Combination of Functions in or uα
Canadian journal of mathematics, Tome 25 (1973) no. 5, pp. 982-985

Voir la notice de l'article provenant de la source Cambridge University Press

Labelle and Rahman [4] showed that if f , g ∈ , the normalized convex functions in the unit disc D, then has a radius of convexity at least as large as the smallest root of 1 – 3r + 2r2 — 2r3 = 0. Their method requires neither the properties of the arithmetic mean nor the strong geometric properties of ; indeed, the procedure works for a linear combination of functions from any linear invariant family of finite order. 
Campbell, Douglas Michael. The Radius of Convexity of a Linear Combination of Functions in or uα. Canadian journal of mathematics, Tome 25 (1973) no. 5, pp. 982-985. doi: 10.4153/CJM-1973-104-5
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[1] 1. Campbell, D. M., Locally univalent functions with locally univalent derivatives, Trans. Amer. Math. Soc. 162 (1971), 395–409. Google Scholar

[2] 2. Campbell, D. M., β-close-to-bounded boundary rotation functions I. (to appear). Google Scholar

[3] 3. Goluzin, G. M., Geometric theory of functions of a complex variable, Amer. Math. Soc. Vol. 26 (Providence, R. I., 1969). Google Scholar

[4] 4. Labelle, G. and Rahman, Q. I., Remarque sur la moyenne arithmétique de fonctions univalentes convexes, Can. J. Math. 21 (1969), 977–981. Google Scholar

[5] 5. Pommerenke, C., Linear-invariante Familien Analytischer Funktionen. I, Math. Ann. 155 (1964), 108–154. Google Scholar

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