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We are finding difficulties in solving this claim:

Let's suppose that $$ \int_a^\infty f(x)^2 dx < \infty \text{ and } \int_a^\infty f''(x)^2 dx < \infty. $$ Prove that $$\int_a^\infty f'(x)^2 dx < \infty.$$

Thank everyone who can solve this claim or give any suggestion.

gt6989b
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michael
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1 Answers1

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Hint: integration by parts and Cauchy-Schwarz.

Robert Israel
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  • we already did it but there is an limit of f(x)*f'(x) after doing integration by parts and how do you proof that this limit exists at all by the way...thank you for the quick response – michael May 03 '16 at 19:19
  • Following this good suggestion by @Robert Israel you see that $\int_a^x (f'(t))^2 , dt = f(x)f'(x) - f(a)f'(a) - \int_a^x f(t)f''(t) , dt$. The integral on the RHS must converge (Cauchy-Schwarz) and since the integrand on the LHS is non-negative, the limit exists -- either finite or $+\infty$.But we can't have $f(x)f'(x) \to +\infty$ because $2(f^2(x) - f^2(a)) = \int_a^x f(t)f'(t) , dt$ and $f^2(x) \to +\infty$ contradicts given that $f^2$ is integrable. – RRL May 04 '16 at 02:31