This commit is contained in:
Bokuan Li
2026-03-06 14:06:15 -05:00
parent 173727665b
commit 5034bc4220
109 changed files with 1184 additions and 410 deletions

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@@ -7,10 +7,12 @@
\[
ab \le \frac{a^p}{p} + \frac{b^q}{q}
\]
and for any $\eps > 0$,
\[
ab \le \eps a^p + \frac{1}{q}(\eps q)^{-q/p}b^q
\]
\end{lemma}
\begin{proof}
Since $x \mapsto \exp(x)$ is convex,
@@ -22,4 +24,5 @@
\[
ab = (\eps p)^{1/p}a \cdot \frac{b}{(\eps p)^{1/p}} \le \eps a^p + \frac{b^q}{q}(\eps p)^{-(1/p)q} = \eps a^p + \frac{1}{q}(\eps q)^{-q/p}b^q
\]
\end{proof}