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

$|\log \frac{z{f}^{\prime }(z)}{f(z)}|\le \log \frac{1+|z|}{1-|z|}.$

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

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

and

$|\arg \frac{z{f}^{\prime }(z)}{f(z)}|\le \log \frac{1+|z|}{1-|z|}.$

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

$g_w(\zeta )=\frac{\zeta }{(1-\bar{w}\zeta {)}^2},$

where |w| = 1.

Proof

Template:Harvtxt originally proved these inequalities based on extremal techniques of Ludwig Bieberbach. Subsequent proofs, outlined in Template:Harvtxt, 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

$|\sum _{i=1}^n\sum _{j=1}^n{\lambda }_i{\lambda }_j\log \frac{g(z_i)-g(z_j)}{z_i-z_j}|\le \sum _{i=1}^n\sum _{j=1}^n{\lambda }_i\overline{{\lambda }_j}\log \frac{z_i\overline{z_j}}{z_i\overline{z_j}-1},$

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

$|\log \frac{g^{\prime }(\zeta )g^{\prime }(\eta )(\zeta -\eta {)}^2}{(g(\zeta )-g(\eta ){)}^2}|\le \log \frac{|1-\zeta \bar{\eta }{|}^2}{(|\zeta {|}^2-1)(|\eta {|}^2-1)}.$

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

$|\log \frac{\zeta g^{\prime }(\zeta )}{g(\zeta )}|\le \frac{|\zeta {|}^2+1}{|\zeta {|}^2-1}.$

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

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

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

$\mathrm{\Re }\frac{z{f}^{\prime }(z)}{f(z)}\ge 0$

for |z| < r. Equivalently

$|\arg \frac{z{f}^{\prime }(z)}{f(z)}|\le \frac{\pi }{2}.$

On the other hand by the inequality of Grunsky above,

$|\arg \frac{z{f}^{\prime }(z)}{f(z)}|\le \log \frac{1+|z|}{1-|z|}.$

Thus if

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

the inequality holds at z. This condition is equivalent to

$|z|\le \tanh \frac{\pi }{4}$

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

References

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