science

Ono's inequality

Ono's inequality 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 Ono's inequality rather than just read about it. In short: In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by Tôda Ono (小野藤太) in 1914, the inequality is actually false; however, the statement is true for acute triangles, as shown by F.

Key takeaways

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

Reference excerpt

In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by Tôda Ono (小野藤太) in 1914, the inequality is actually false; however, the statement is true for acute triangles, as shown by F. Balitrand in 1916.

Statement of the inequality Consider an acute triangle (meaning a triangle with three acute angles) in the Euclidean plane with side lengths a, b and c and area S. Then

27 ( b 2 + c 2 − a 2 ) 2 ( c 2 + a 2 − b 2 ) 2 ( a 2 + b 2 − c 2 ) 2 ≤ ( 4 S ) 6 . {\displaystyle 27(b^{2}+c^{2}-a^{2})^{2}(c^{2}+a^{2}-b^{2})^{2}(a^{2}+b^{2}-c^{2})^{2}\leq (4S)^{6}.}

This inequality fails for general triangles (to which Ono's original conjecture applied), as shown by the counterexample a = 2 , b = 3 , c = 4 , S = 3 15 / 4. {\displaystyle a=2,\,\,b=3,\,\,c=4,\,\,S=3{\sqrt {15}}/4.}

The inequality holds with equality in the case of an equilateral triangle, in which up to similarity we have sides 1 , 1 , 1 {\displaystyle 1,1,1} and area 3 / 4. {\displaystyle {\sqrt {3}}/4.}

Proof Dividing both sides of the inequality by 64 ( a b c ) 4 {\displaystyle 64(abc)^{4}} , we obtain:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Ono's inequality

Start with the simplest possible case. Write down what Ono's inequality 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 Ono's inequality 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 Ono's inequality 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 Ono's inequality

In research: Ono's inequality 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 Ono's inequality 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: Ono's inequality is common in secondary-school and first-year university syllabi. It links to neighbouring topics Disproved conjectures, Triangle inequalities, so understanding it makes those chapters shorter.
In everyday life: Look for Ono's inequality 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 Ono's inequality in 20 minutes

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

Frequently asked questions

What is Ono's inequality in simple terms?

Why does Ono's inequality 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 Ono's inequality?

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 Ono's inequality.