Inequalities path · I07

Before this lesson: Arithmetic, geometric and harmonic means, Weighted means, Hölder’s inequality

Your goal: Choose a useful pair of means with valid inputs and exponents.

Start with the idea and one example. Take a break before the written task if you need it. The questions are original teaching exercises, not official past-paper questions.

Names you may know: generalised mean inequality; RMS-AM-GM-HM.

Power mean inequality: the key idea

For positive x₁,…,xₙ and real r≠0, the power mean is Mr=(x1r+⋯+xnrn)1/rM_r=\left(\frac{x_1^r+\cdots+x_n^r}{n}\right)^{1/r}. Define M₀ as the geometric mean, the limit as r tends to 0. The power mean theorem states Mᵣ≥Mₛ when r>s; equality for positive inputs occurs exactly when all inputs agree. The familiar chain is root-mean-square M₂ ≥ arithmetic M₁ ≥ geometric M₀ ≥ harmonic M₋₁. Negative orders require nonzero positive inputs. This lesson uses the theorem and proves its quadratic special case; a full proof for all real orders requires an additional convexity argument.

A worked example

Find the arithmetic mean and root-mean-square of 1 and 7.

The arithmetic mean is 4. The RMS is 1+492=5\sqrt{\frac{1+49}{2}}=5. Thus 5≥4; equality fails because the inputs differ.

Your turn: change one thing

For positive a,b show a²+b²≥(a+b)²/2.

Try this on paper before opening the explanation.

Compare your reasoning

Twice the left side minus (a+b)² is (a−b)²≥0. Taking positive square roots gives RMS≥AM.

Pause and check

A trap to avoid: Overlooking zero inputs when exponents are negative.

Practise and adjust the level

Foundation checks the language; Core applies the method; Stretch asks you to choose or justify an idea. These are levels within this lesson. A session selects six of the nine questions; unused questions allow the level to change. Advanced theory still needs written practice.

Interactive practice loads here. You can also use the complete question set below.

Write a complete argument

Prove RMS≥AM for n positive inputs.

Planning hint

List the assumptions and the exact conclusion. Identify the definition or theorem in this lesson that connects them. Explain why its conditions hold before using it.

Read the full solution after your attempt

Write μ=∑xin\mu=\frac{\sum x_i}{n}. Since ∑(xi−μ)2=∑xi2−nμ2≥0\sum (x_{i}-\mu )^{2}=\sum x_{i}^{2}-n\mu ^{2}\ge 0, we have ∑xi2n≥μ2\frac{\sum x_i^2}{n}\ge\mu^2. Both sides have nonnegative square roots, giving RMS≥AM. Equality means every xᵢ=μ.

My proof notebook

Write on paper, or keep a draft here. Compare your reasoning with the solution only after a real attempt. The checklist is your own review, not an automatic mark.

All nine practice questions

Prefer paper or have JavaScript switched off? The complete question set, hints and solutions are here. Interactive practice uses these same questions in an order chosen from your answers.

1. Which power mean is the arithmetic mean?

Foundation

  1. M₁
  2. M₂
  3. M₀
  4. M₋₁
Hint

Set the exponent to 1.

Answer and reasoning

M₁. M₁ is the sum divided by n.

2. Which mean is M₀ by definition?

Foundation

  1. Geometric
  2. Arithmetic
  3. Harmonic
  4. Maximum
Hint

Use the limiting case.

Answer and reasoning

Geometric. M₀ is defined as the geometric mean.

3. What is the RMS of 1 and 7?

Core

  1. 8
  2. 25
  3. 5
  4. 4
Hint

Average the squares, then take a square root.

Answer and reasoning

5. 1+492=25=5\sqrt{\frac{1+49}{2}}=\sqrt{25}=5.

4. For positive inputs, which comparison is valid?

Core

  1. M₃=M₁ always
  2. M₃<0
  3. M₃≥M₁
  4. M₃≤M₁ always
Hint

Larger order gives a larger mean.

Answer and reasoning

M₃≥M₁. The power mean theorem applies since 3>1.

5. When are M₂ and M₁ equal?

Stretch

  1. Their sum is 1
  2. There are two inputs
  3. All inputs are integers
  4. All inputs are equal
Hint

Use the variance identity.

Answer and reasoning

All inputs are equal. The sum of squared deviations vanishes exactly when every input equals the mean.

6. Why exclude zero inputs when using M₋₁?

Stretch

  1. Zero is negative
  2. Means cannot be zero
  3. The order must be integral
  4. Reciprocals of zero are undefined
Hint

Inspect x⁻¹.

Answer and reasoning

Reciprocals of zero are undefined. The harmonic mean involves 1/x for every input.

7. The root-mean-square is which power mean?

Foundation

  1. M₁
  2. M₀
  3. M₋₁
  4. M₂
Hint

The inputs are squared before averaging.

Answer and reasoning

M₂. Its exponent parameter is 2.

8. What is the RMS of 3 and 3?

Core

  1. 3
  2. 6
  3. 9
  4. 3\sqrt{3}
Hint

Average equal squares and take the square root.

Answer and reasoning

3. 9+92=3\sqrt{\frac{9+9}{2}}=3.

9. For unequal positive inputs and r>s, how do Mᵣ and Mₛ compare?

Stretch

  1. Mᵣ>Mₛ
  2. They are equal
  3. Mᵣ<Mₛ
  4. They are both zero
Hint

Equality requires every input equal.

Answer and reasoning

Mᵣ>Mₛ. The power-mean inequality is strict for unequal positive inputs.

Choose your next step

Continue to Rearrangement inequality. If this felt difficult, return to a prerequisite above. Every lesson stays open.

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