mathematics

Hardy's theorem

Hardy'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 Hardy's theorem rather than just read about it. In short: In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let f {\displaystyle f} be a holomorphic function on the open ball centered at zero and radius R {\displaystyle R} in the complex plane, and assume that f {\displaystyle f} is not a constant function.

Key takeaways

Hardy'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 Hardy's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hardy's theorem from memory before moving on to harder problems.

Reference excerpt

In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let f {\displaystyle f} be a holomorphic function on the open ball centered at zero and radius R {\displaystyle R} in the complex plane, and assume that f {\displaystyle f} is not a constant function. If one defines

I ( r ) = 1 2 π ∫ 0 2 π | f ( r e i θ ) | d θ {\displaystyle I(r)={\frac {1}{2\pi }}\int _{0}^{2\pi }\!\left|f(re^{i\theta })\right|\,d\theta }

for 0 < r < R , {\displaystyle 0<r<R,} then this function is strictly increasing and is a convex function of log ⁡ r {\displaystyle \log r} .

See also Maximum principle Hadamard three-circle theorem

References John B. Conway. (1978) Functions of One Complex Variable I. Springer-Verlag, New York, New York. This article incorporates material from Hardy's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Worked examples

Example 1 — a first encounter with Hardy's theorem

Start with the simplest possible case. Write down what Hardy'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 Hardy'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 Hardy'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 Hardy's theorem

In research: Hardy'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 Hardy'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: Hardy'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 Hardy'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.

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How to study Hardy's theorem in 20 minutes

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

Frequently asked questions

What is Hardy's theorem in simple terms?

Why does Hardy'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 Hardy'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 Hardy's theorem.