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

:$\left, \log \\le \log .$

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

:$\left, \\le$

and

:$\left, \arg \ \le \log .$

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

:$g_w(\zeta)=,$

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

:$g(z) = z + b_1 z^ + b_2 z^ + \cdots.$

Goluzin's inequalities state that

:$\left, \sum_^n\sum_^n\lambda_i\lambda_j \log \ \le \sum_^n\sum_^n \lambda_i\overline\log ,$

where the ''z''_''i'' are distinct points with , ''z''_''i'', > 1 and λ_''i'' are arbitrary complex numbers.

Taking ''n'' = 2. with λ_''1'' = – λ_''2'' = λ, the inequality implies

:$\left, \log \\le \log .$

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

:$\left, \log \ \le .$

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

:$g(\zeta)=f(\zeta^)^$

with $z=\zeta^.$

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

:$\Re \ge 0$

for '', z, '' < ''r''. Equivalently

:$\left, \arg \ \le .$

On the other hand by the inequality of Grunsky above,

:$\left, \arg \\le \log .$

Thus if

:$\log \le ,$

the inequality holds at ''z''. This condition is equivalent to

:$, z, \le \tanh$

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

References

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*{{citation, last=Pommerenke, first= C., authorlink=Christian Pommerenke, title=Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, series= Studia Mathematica/Mathematische Lehrbücher, volume=15, publisher= Vandenhoeck & Ruprecht, year= 1975

Theorems in complex analysis