Assume that $f$ is a real continuous function on $[0,1]$. I need the smallest constant $C$ in the following inequality $$4\int_0^1\left(\int_0^\rho r f(r)dr\right)^2\frac{d\rho}{\rho}+\left(\int_0^1 r f(r)dr\right)^2\le C \int_0^1 r f^2(r)dr.$$
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4$\begingroup$ Is there some context for where this comes form? Do you need the precise best constant and do you have reason to suppose it will have a nice closed form? $\endgroup$– Sam ZbarskyCommented Jun 25, 2020 at 23:52
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$\begingroup$ It has to do with the norm of some operator, but complicated to explain... $\endgroup$– LiraCommented Jun 26, 2020 at 15:37
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1 Answer
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$C=1.0986702957852257$ is the solution. It can be proved by using the Langrange multipliers.
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$\begingroup$ If you are satisfied with this answer, then you should accept it. $\endgroup$– LSpiceCommented Nov 29, 2021 at 1:59