If $f\in C^1[0,1)$ and $f(0)=0$ , show that $\int_0^1\frac{|f(x)|^2}{x^2}dx\leq4\int_0^1|f^{'}(x)|^2dx$
The book solves the problem using variational method. But I want to seek for an elementary solution for it. Any idea?
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It seems that it is an special condition of famous Hardy's inequality. – L'enfer Apr 24 '22 at 17:07
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See here https://math.stackexchange.com/q/126889/121671. This may provide the solution you are looking for by match the right terms. – Mittens Apr 24 '22 at 17:57
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Thanks! @Oliver Díaz – L'enfer Apr 25 '22 at 02:02