Inmathematics,Grunsky's theorem, due to the German mathematicianHelmut Grunsky, is a result incomplex analysis concerningholomorphicunivalent functions defined on theunit disk in thecomplex numbers. The theorem states that a univalent function defined on the unit disc, fixing the point 0, maps every disk|z| <r onto astarlike domain forr ≤ tanh π/4. The largestr for which this is true is called theradius of starlikeness of the function.

Letf be a univalent holomorphic function on the unit discD such thatf(0) = 0. Then for allr ≤ tanh π/4, the image of the disc|z| <r isstarlike with respect to 0, , i.e. it is invariant under multiplication by real numbers in (0,1).

Iff(z) is univalent onD withf(0) = 0, then

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

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

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

and

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

For fixedz, both these equalities are attained by suitableKoebe functions

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

where|w| = 1.

Grunsky (1932) originally proved these inequalities based on extremal techniques ofLudwig Bieberbach. Subsequent proofs, outlined inGoluzin (1939), relied on theLoewner equation. More elementary proofs were subsequently given based onGoluzin's inequalities, an equivalent form of Grunsky's inequalities (1939) for theGrunsky matrix.

For a univalent functiong inz > 1 with an expansion

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

Goluzin's inequalities state that

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

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

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

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

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

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

Finally iff is any normalized univalent function inD, the required inequality forf follows by taking

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

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

Letf be a univalent function onD withf(0) = 0. ByNevanlinna's criterion,f is starlike on|z| <r if and only if

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

for|z| <r. Equivalently

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

On the other hand by the inequality of Grunsky above,

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

Thus if

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

the inequality holds atz. This condition is equivalent to

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

and hencef is starlike on any disk|z| <r withr ≤ tanh π/4.

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• Theorems in complex analysis