Inequality Proof Problems [401-450]
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P401. Let $a_1, a_2, \ldots, a_n$ be positive real numbers with
\[a_1 \ge a_2 \ge \cdots \ge a_n.\]Show that
\[\left(\frac{a_1+a_2}{2}\right)\left(\frac{a_2+a_3}{2}\right)\cdots\left(\frac{a_n+a_1}{2}\right) \le \left(\frac{a_1+a_2+a_3}{3}\right)\left(\frac{a_2+a_3+a_4}{3}\right)\cdots\left(\frac{a_n+a_1+a_2}{3}\right).\]A401.
P402. Autocorrelation Inequality
Let \(f \ge 0\) and suppose the support of \(f\) is contained in \([-\tfrac14,\tfrac14]\). Assume that, for all such functions \(f\), the following inequality holds:
\[\max_{|t|\le \tfrac12} (f * f)(t) \ge C_1\left(\int f\right)^2.\]Find an upper bound for \(C_1\). Note: This is essentially a “min–max” problem (minimizing the worst-case / largest value).
A402. The best currently known upper bound is
\[C_1 \le 1.50286.\]See ref. Future progress may further lower this upper bound.
P403. Autocorrelation Inequality
Let \(f \ge 0\). Define
\[C_2=\sup_{f\ge 0}\frac{\|f*f\|_2^2}{\|f*f\|_1\,\|f*f\|_\infty}.\]Find a lower bound for \(C_2\). Note: This is essentially about achieving a large value (maximization via the supremum).
A403. The best currently known lower bound is
\[C_2 \ge 0.9610.\]See ref). Future progress may further raise this lower bound.
Input: 2026.01.22 22:08