Proving Jensen's Inequality

★ 7/10functions Convexity Induction inequality

Problem

Let $f(x)$ be defined on an interval $I$ , and suppose that for all $x_1, x_2 \in I$ and all $t$ with $0 \leqslant t \leqslant 1$ ,

f\big((1-t)x_1 + tx_2\big) \leqslant (1-t)f(x_1) + t f(x_2).

Let $n \geqslant 2$ , let $x_i \in I$ for $1 \leqslant i \leqslant n$ , and let $p_i \geqslant 0$ with $\displaystyle\sum_{i=1}^{n} p_i = 1$ . Prove that

f\!\left(\sum_{i=1}^{n} p_i x_i\right) \leqslant \sum_{i=1}^{n} p_i f(x_i).

Answer

Solution

Difficulty	7/10
Topics	functions, Convexity, Induction, inequality

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