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If $f$ is continuous and $3\geq f(x)\geq 1$ for all $ x\in [0,1]$, show that $$ 1 \leq \int_0^1f(x)dx\int_0^1\Bigg(\frac{1}{f(x)}\Bigg)dx \leq \frac{4}{3}. $$

Thanks!

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By C-S $$ \int\limits_0^1f(x)dx\int\limits_0^1\frac{1}{f(x)}dx \geq \left(\int\limits_0^11dx\right)^2=1.$$ For the proof of the right inequality use the Schweitzer's inequality.

See here:

http://www.ssmrmh.ro/wp-content/uploads/2016/08/INTEGRAL-FORMS-FOR-SCHWEITZERS-AND-POLYA-SZEGOS-INEQUALITIES.pdf

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