i proved that $-log(x)\in L^p((0,1))$ for $1<p<\infty$, but how can i prove that it's true for $0<p<1$ too? Any suggestion or hint? Thank you!!
2026-05-05 20:28:03.1778012883
$-log(x) \in L^p((0,1))$ for $0<p<1$
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Note that $|\log x|\le 1$ on $[1/e,1].$ Hence $|\log x|^p\le 1$ on $[1/e,1]$ for $0<p<\infty.$ So we need worry only about what happens on $(0,1/e],$ where $|\log x|> 1.$ You have shown $|\log x|\in L^2(0,1/e].$ Since $|\log x|^p \le |\log x|^2$ on this interval for $0<p<2,$ we're done.