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

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

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

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

and

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

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

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

where |w| = 1.

Proof

Grunsky (1932) originally proved these inequalities based on extremal techniques of Ludwig Bieberbach. Subsequent proofs, outlined in 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

${\displaystyle g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots .}$

Goluzin's inequalities state that

${\displaystyle \left|\sum _{i=1}^{n}\sum _{j=1}^{n}\lambda _{i}\lambda _{j}\log {g(z_{i})-g(z_{j}) \over z_{i}-z_{j}}\right|\leq \sum _{i=1}^{n}\sum _{j=1}^{n}\lambda _{i}{\overline {\lambda _{j}}}\log {z_{i}{\overline {z_{j}}} \over z_{i}{\overline {z_{j}}}-1},}$

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

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

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

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

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

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

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

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

${\displaystyle \Re {zf^{\prime }(z) \over f(z)}\geq 0}$

for |z| < r. Equivalently

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

On the other hand by the inequality of Grunsky above,

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

Thus if

${\displaystyle \log {1+|z| \over 1-|z|}\leq {\pi \over 2},}$

the inequality holds at z. This condition is equivalent to

${\displaystyle |z|\leq \tanh {\pi \over 4}}$

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

References

• Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, pp. 95–98, ISBN 0-387-90795-5
• Goluzin, G.M. (1939), "Interior problems of the theory of univalent functions", Uspekhi Mat. Nauk, 6: 26–89 (in Russian)
• Goluzin, G. M. (1969), Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, vol. 26, American Mathematical Society
• Goodman, A.W. (1983), Univalent functions, vol. I, Mariner Publishing Co., ISBN 0-936166-10-X
• Goodman, A.W. (1983), Univalent functions, vol. II, Mariner Publishing Co., ISBN 0-936166-11-8
• Grunsky, H. (1932), "Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Bereiche (inaugural dissertation)", Schr. Math. Inst. U. Inst. Angew. Math. Univ. Berlin, 1: 95–140, archived from the original on 2015-02-11, retrieved 2011-12-07 (in German)
• Grunsky, H. (1934), "Zwei Bemerkungen zur konformen Abbildung", Jber. Deutsch. Math.-Verein., 43: 140–143 (in German)
• Hayman, W. K. (1994), Multivalent functions, Cambridge Tracts in Mathematics, vol. 110 (2nd ed.), Cambridge University Press, ISBN 0-521-46026-3
• Nevanlinna, R. (1921), "Über die konforme Abbildung von Sterngebieten", Öfvers. Finska Vet. Soc. Forh., 53: 1–21
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht