Geodesic
Estimates for the Koebe Constant and the Second Coefficient for Some Classes of Univalent Functions
Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1311-1324

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

Let S be the set of all normalized univalent analytic functions ƒ(z) = z + a2z2 + ... in the open unit disk U. Then ƒ(U) contains the disk . Here is the best possible constant and is referred to as the Koebe constant for S. On the other extreme, ƒ(U) cannot contain the disk {|w| < 1}; unless ƒ is the identity mapping.In order to interpolate between the class S and the identity mapping, one may introduce the families , of functions ƒ ∈ S such that ƒ(U) contains the disk {|w| < d};. Then S(d1) ⊃ S(d2) for d1 < d2, and S(1) contains only the identity mapping. It is obvious that d is the “Koebe constant” for S(d). The relation between d and the second coefficient a 2 has been studied by E. Netanyahu [5, 6]. 
Bshouty, D.; Hengartner, W.; Schober, G. Estimates for the Koebe Constant and the Second Coefficient for Some Classes of Univalent Functions. Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1311-1324. doi: 10.4153/CJM-1980-101-9
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