science

Goodman's conjecture

Goodman's conjecture is a science 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 Goodman's conjecture rather than just read about it. In short: Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician. Formulation Let f ( z ) = ∑ n = 1 ∞ b n z n {\displaystyle f(z)=\sum _{n=1}^{\infty }{b_{n}z^{n}}} be a p {\displaystyle p} -valent function.

Key takeaways

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

Reference excerpt

Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician.

Formulation Let f ( z ) = ∑ n = 1 ∞ b n z n {\displaystyle f(z)=\sum _{n=1}^{\infty }{b_{n}z^{n}}} be a p {\displaystyle p} -valent function. The conjecture claims the following coefficients hold:

| b n | ≤ ∑ k = 1 p 2 k ( n + p ) ! ( p − k ) ! ( p + k ) ! ( n − p − 1 ) ! ( n 2 − k 2 ) | b k | {\displaystyle |b_{n}|\leq \sum _{k=1}^{p}{\frac {2k(n+p)!}{(p-k)!(p+k)!(n-p-1)!(n^{2}-k^{2})}}|b_{k}|}

Partial results It's known that when p = 2 , 3 {\displaystyle p=2,3} , the conjecture is true for functions of the form P ∘ ϕ {\displaystyle P\circ \phi } where P {\displaystyle P} is a polynomial and ϕ {\displaystyle \phi } is univalent.

External sources Goodman, A. W. (1948). "On some determinants related to 𝑝-valent functions". Transactions of the American Mathematical Society. 63: 175–192. doi:10.1090/S0002-9947-1948-0023910-X. Lyzzaik, Abdallah; Styer, David (1978). "Goodman's conjecture and the coefficients of univalent functions". Proceedings of the American Mathematical Society. 69: 111–114. doi:10.1090/S0002-9939-1978-0460619-7. Grinshpan, Arcadii Z. (2002). "Logarithmic Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains". Geometric Function Theory. Handbook of Complex Analysis. Vol. 1. pp. 273–332. doi:10.1016/S1874-5709(02)80012-9. ISBN 978-0-444-82845-3. AGrinshpan, A.Z. (1997). "On the Goodman conjecture and related functions of several complex variables". Department of Mathematics, University of South Florida, Tampa, FL. 9 (3): 198–204. MR 1466800. Grinshpan, A. Z. (1995). "On an identity related to multivalent functions". Proceedings of the American Mathematical Society. 123 (4): 1199. doi:10.1090/S0002-9939-1995-1242085-7.

Worked examples

Example 1 — a first encounter with Goodman's conjecture

Start with the simplest possible case. Write down what Goodman's conjecture claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Goodman's conjecture 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 Goodman's conjecture 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 Goodman's conjecture

In research: Goodman's conjecture appears in science 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 Goodman's conjecture 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: Goodman's conjecture is common in secondary-school and first-year university syllabi. It links to neighbouring topics Complex analysis, Conjectures, so understanding it makes those chapters shorter.
In everyday life: Look for Goodman's conjecture 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.

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How to study Goodman's conjecture in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Goodman's conjecture 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 Goodman's conjecture out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Goodman's conjecture in simple terms?

Goodman's conjecture on the coefficients of multivalent functions was proposed in complex analysis in 1948 by Adolph Winkler Goodman, an American mathematician. Formulation Let f ( z ) = ∑ n = 1 ∞ b n z n {\displaystyle f(z)=\sum _{n=1}^{\infty }{b_{n}z^{n}}} be a p {\displaystyle p} -valent functi…

Why does Goodman's conjecture matter?

Because it connects several science 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 Goodman's conjecture?

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 Goodman's conjecture.