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

: $\left|\log {zf^\prime(z)\over f(z)}\right|\le \log {1+|z|\over 1-|z|}.$

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

: $\left|{zf^\prime(z)\over f(z)}\right|\le {1+|z|\over 1-|z|}$

and

: $\left|\arg {zf^\prime(z)\over f(z)}\right| \le \log {1+|z|\over 1-|z|}.$

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

: $g_w(\zeta)={\zeta\over (1-\overline{w}\zeta)^2},$

where |w| = 1.

=Proof=

{{harvtxt|Grunsky|1932}} originally proved these inequalities based on extremal techniques of Ludwig Bieberbach. Subsequent proofs, outlined in {{harvtxt|Goluzin|1939}}, 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

: $g(z) = z + b_1 z^{-1} + b_2 z^{-2} + \cdots.$

Goluzin's inequalities state that

: $\left|\sum_{i=1}^n\sum_{j=1}^n\lambda_i\lambda_j \log {g(z_i)-g(z_j)\over z_i-z_j}\right| \le \sum_{i=1}^n\sum_{j=1}^n \lambda_i\overline{\lambda_j}\log {z_i\overline{z_j}\over z_i\overline{z_j}-1},$

where the z_i are distinct points with |z_i| > 1 and λ_i are arbitrary complex numbers.

Taking n = 2. with λ₁ = – λ₂ = λ, the inequality implies

: $\left| \log {g^\prime(\zeta)g^\prime(\eta) (\zeta-\eta)^2\over (g(\zeta)-g(\eta))^2}\right|\le \log {|1-\zeta\overline{\eta}|^2\over (|\zeta|^2 -1 )(|\eta|^2 -1)}.$

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

: $\left| \log {\zeta g^\prime(\zeta) \over g(\zeta)}\right| \le {|\zeta|^2 + 1\over |\zeta|^2 -1}.$

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

: $g(\zeta)=f(\zeta^{-2})^{-{1\over 2}}$

with $z=\zeta^{-2}.$

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

: $\Re {zf^\prime(z)\over f(z)} \ge 0$

for |z| < r. Equivalently

: $\left|\arg {zf^\prime(z)\over f(z)}\right| \le {\pi\over 2}.$

On the other hand, by the inequality of Grunsky above,

: $\left|\arg {zf^\prime(z)\over f(z)}\right|\le \log {1+|z|\over 1-|z|}.$

Thus if

: $\log {1+|z|\over 1-|z|} \le {\pi\over 2},$

the inequality holds at z. This condition is equivalent to

: $|z|\le \tanh {\pi\over 4}$

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

References

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Category:Theorems in complex analysis