Jensen's Inequality and Convex Functions: Algebra Notes for IMO (Maths Olympiad)

📚 Study Notes – Jensen’s Inequality & Convex Functions

(Designed for beginners, no heavy jargon – just clear ideas and tiny examples.)

1️⃣ What is a Convex Function?

Think of a bowl – the sides curve upward, the bottom is flat or slightly curved.
If you put a straight stick (a line segment) between any two points on the bowl’s surface, the stick will always sit above the bowl.
That’s what “convex” means.

Every point on the curve looks like a “shelf” that never dips below the straight line between two points.
If the curve ever dips below that straight line, it is not convex.

2️⃣ How to Spot a Convex Function

You can use two quick tools:

Tool	What to do	Why it works
Second derivative test	Find f''(x). If f''(x) ≥ 0 everywhere, the curve is convex.	The second derivative tells you how the slope changes; non‑negative means the slope never decreases.
Graph sketch	Draw the curve. Pick two points, draw a line between them, and check if the curve stays below that line.	Visual check; good for simple functions.

Tip: For a quadratic f(x) = ax² + bx + c, the graph is always a parabola.
If a > 0, it opens upward → convex.
If a < 0, it opens downward → not convex.

3️⃣ Jensen’s Inequality – The Big Idea

Jensen’s Inequality tells us how a convex function behaves with averages.

Simple statement
If you take the average of some numbers and then plug that average into a convex function, the result is at least the same as if you had plugged each number into the function first and then averaged those results.

Mathematically (but we’ll keep it intuitive):

f(average of x's)  ≥  average of f(x)'s

4️⃣ Everyday Analogy

Imagine you have several identical bowl-shaped containers (the convex function).
If you pour the same amount of water into each container and then mix the water together, the total height of the mixed water is not lower than if you had mixed the containers’ contents first and then poured them into a single bowl.

In short: “Average first, then shape, gives a higher or equal result.”

5️⃣ Tiny Example

Let’s use the simple convex function f(x) = x².

Choose numbers: 1, 3, and 5.
- Their average: (1+3+5)/3 = 3.
- Apply f to the average: f(3) = 9.
Apply f first, then average:
- f(1)=1, f(3)=9, f(5)=25.
- Their average: (1+9+25)/3 = 35/3 ≈ 11.67.
Compare:
- 9 (average→f) ≤ 11.67 (f→average).
- The inequality holds.

6️⃣ Why This Matters in IMO Problems

Bounding sums and products: Jensen lets you replace a messy sum by a simpler expression.
Proving inequalities: Instead of juggling many terms, show the function is convex and apply Jensen.
Handling averages: Many Olympiad problems ask for the minimum or maximum of a symmetric expression; Jensen often gives the answer directly.

7️⃣ Quick Checklist to Use Jensen

Identify the function you want to apply.
Check convexity (second derivative ≥ 0 or graph test).
Write the numbers you’re averaging (weights can be equal or different).
Apply Jensen:
- Left side: f(weighted average)
- Right side: weighted average of f(values)
Interpret the result in the context of the problem.

8️⃣ Common Pitfalls

Mistake	Fix
Assuming every function is convex	Always test convexity first.
Forgetting that weights must sum to 1	Normalize the weights.
Mixing up “average first” vs. “apply first”	Keep the two sides of the inequality clear.
Ignoring that equality happens only when all numbers are equal	Remember Jensen can be strict.

9️⃣ Mini Practice Problems

Prove that for any positive numbers a, b, c:
a² + b² + c² ≥ 3( (a+b+c)/3 )².
(Hint: use f(x)=x²)
Find the minimum of x + 1/x for x > 0.
(Think about convexity of f(x)=x + 1/x)
Show that for any real numbers x₁, x₂, …, x_n:
∑ ln(1 + x_i) ≤ n ln(1 + (∑ x_i)/n) if x_i > -1.
(Hint: use f(x)=ln(1+x), which is concave; remember Jensen works the other way for concave functions.)

🔚 Summary

Convex function: Bowl-shaped curve; line segment stays above it.
Convex test: