Grunsky's theorem
In mathematics, Grunsky's theorem, due to the German mathematician Helmut Grunsky, is a result in complex analysis concerning holomorphic univalent functions defined on the unit disk in the complex numbers. The theorem  states that a univalent function defined on the unit disc, fixing the point 0, maps every disk |z| < r onto a starlike domain for r ≤ tanh π/4. The largest r for which this is true is called the radius of starlikeness of the function. Statement 
Let f be a univalent holomorphic function on the unit disc D such that f(0) = 0. Then for all r ≤ tanh π/4, the image of the disc |z| < r is starlike with respect to 0, , i.e. it is invariant under multiplication by real numbers in (0,1). An inequality of Grunsky
If f(z) is univalent on D with f(0) = 0, then

Taking the real and imaginary parts of the logarithm, this implies the two inequalities

and

For fixed z, both these equalities are attained by suitable Koebe functions

where |w| = 1. Proof
 originally proved these inequalities based on extremal techniques of Ludwig Bieberbach. Subsequent proofs, outlined in , relied on the Loewner equation. More elementary proofs were subsequently given based on Goluzin's inequalities, an equivalent form of Grunsky's inequalities (1939) for the Grunsky matrix. For a univalent function g  in z > 1 with an expansion

Goluzin's inequalities state that

where the zi  are distinct points with |zi| > 1  and λi are arbitrary complex numbers. Taking n = 2. with λ1 = – λ2 = λ, the inequality implies

If g is an odd function and  η =  – ζ, this yields

Finally if f is any normalized univalent function in D, the required inequality for f follows by taking

with

Proof of the theorem
Let f be a univalent function on D with f(0) = 0. By Nevanlinna's criterion,  f is starlike on |z| < r if and only if

for |z| < r. Equivalently

On the other hand by the inequality of Grunsky above,

Thus if

the inequality holds at z. This condition is equivalent to

and hence f is starlike on any disk |z| < r with r ≤ tanh π/4. References

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Theorems in complex analysis