A Carleman-Style Estimate

★ 8/10algebra AM-GM Telescoping inequality

Problem

Let $a_i, b_i > 0$ for $1 \leqslant i \leqslant n+1$ , and suppose $b_{i+1} - b_i \geqslant \delta > 0$ for $i = 1, 2, \dots, n$ , where $\delta$ is a constant. If $\displaystyle\sum_{i=1}^{n} a_i = 1$ , prove that

\sum_{i=1}^{n} \frac{\sqrt[i]{a_1 a_2 \cdots a_i\, b_1 b_2 \cdots b_i}}{b_{i+1} b_i} < \frac{1}{\delta}.

Answer

Solution

Difficulty	8/10
Topics	algebra, AM-GM, Telescoping, inequality

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