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Published online by Cambridge University Press: 09 April 2009
Bieberbach's conjecture, proposed in 1916 and still unsolved, states that if ƒ(z) = z+a2z2+… is holomorphic and univalent in the disc ∣z∣ < 1 then ∣an∣ ≦ n for each n ≧ 2, with equality for some n only if ƒ(z) is the Koebe function 
of is obtained from this function by a rotation. Very recently Bombieri has succeeded in showing that if ƒ(z) is sufficiently close to the Koebe function, then 
with equality only if ƒ(z) = k(z). This had previously been proved by Garabedian, Ross and Schiffer [3] for even values of n.
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