Inequalities Flashcards

Which classical inequality states that for positive reals a and b, (a + b) / 2 ≥ √(ab)?

Answer: AM-GM inequality

The AM-GM inequality (Arithmetic Mean – Geometric Mean) states that the arithmetic mean of non-negative reals is always at least their geometric mean.

For positive reals a and b, what is the minimum value of a/b + b/a?

Answer: 2

By AM-GM, a/b + b/a ≥ 2√(a/b · b/a) = 2, with equality when a = b.

The Cauchy-Schwarz inequality for real sequences a₁, …, aₙ and b₁, …, bₙ states:

Answer: (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²)

The Cauchy-Schwarz inequality states (Σaᵢbᵢ)² ≤ (Σaᵢ²)(Σbᵢ²), with equality when the sequences are proportional.

For a convex function f and real numbers x, y, Jensen's inequality gives:

Answer: f((x+y)/2) ≤ (f(x)+f(y))/2

For a convex function, Jensen's inequality states f((x+y)/2) ≤ (f(x)+f(y))/2, reflecting that the function lies below the chord between two points.

The inequality a/(b+c) + b/(a+c) + c/(a+b) ≥ 3/2 for positive reals a, b, c is known as:

Answer: Nesbitt's inequality

Nesbitt's inequality states that for positive reals a, b, c, the cyclic sum a/(b+c) + b/(a+c) + c/(a+b) is always at least 3/2.

If x > 0, the minimum value of x + 1/x is:

Answer: 2

By AM-GM, x + 1/x ≥ 2√(x · 1/x) = 2, with equality at x = 1.

For positive reals a, b, c with a + b + c = 3, what is the minimum value of a² + b² + c²?

Answer: 3

By the Cauchy-Schwarz or QM-AM inequality, a² + b² + c² ≥ (a+b+c)²/3 = 3, with equality when a = b = c = 1.