mathematics

Grunsky's theorem

Grunsky's theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Grunsky's theorem rather than just read about it. In short: 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.

Key takeaways

Grunsky's theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Grunsky's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Grunsky's theorem from memory before moving on to harder problems.

Reference excerpt

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 radius of starlikeness of an univalent function f satisfying f(0) = 0 is the largest radius r for which the function f maps the open disk |z| < r into a starlike domain with respect to the origin.

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 closed 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 ⁡ z f ′ ( z ) f ( z ) | ≤ log ⁡ 1 + | z | 1 − | z | . {\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

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

and

| arg ⁡ z f ′ ( z ) f ( z ) | ≤ log ⁡ 1 + | z | 1 − | z | . {\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

g w ( ζ ) = ζ ( 1 − w ¯ ζ ) 2 , {\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

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

Goluzin's inequalities state that

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Grunsky's theorem

Start with the simplest possible case. Write down what Grunsky's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Grunsky's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Grunsky's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Grunsky's theorem

In research: Grunsky's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Grunsky's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Grunsky's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems in complex analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Grunsky's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Grunsky's theorem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Grunsky's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Grunsky's theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Grunsky's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Grunsky's theorem in simple terms?

Why does Grunsky's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Grunsky's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Grunsky's theorem.