Fun of Thinking — Math Problems from Around the World

Convexity and Jensen

Problem

Call $f(x)$ convex on $(a, b)$ if it is differentiable there and $f'(x)$ is increasing.

1. Decide whether $y = x^3$ and $y = \ln\dfrac{1}{x}$ are convex on their domains. 2. Let $f(x)$ be convex on $(a, b)$ . Prove that if $\lambda_1 + \lambda_2 + \cdots + \lambda_n = 1$ with all $\lambda_i > 0$ , then for any $x_i \in (a, b)$ ,

\lambda_1f(x_1) + \lambda_2f(x_2) + \cdots + \lambda_nf(x_n) \geqslant f\left(\lambda_1x_1 + \lambda_2x_2 + \cdots + \lambda_nx_n\right).

For $a, b, c > 0$ and integer $n \geqslant 6$ , prove that

a^n + b^n + c^n \geqslant a^{n-5}b^3c^2 + b^{n-5}c^3a^2 + c^{n-5}a^3b^2.

Answer

Solution

Difficulty	9/10
Topics	calculus, Convexity, Monotonicity, AM-GM, inequality

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