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Where could I learn techniques for solving integral inequalities such as

Given $\int_{\frac13}^{\frac23}f(x)dx=0$, how to prove $4860(\int_0^1f(x)dx)^2\le 11\int_0^1|f''(x)|^2dx$?

Prove that $\left|30240\int_{0}^{1}x(1-x)f(x)f'(x)dx\right|\le1$. ?

One of the posts linked above mentioned polynomial interpolation as a possible approach. Where could I found out more about this idea?

Where do these inequalities come from in general? Many thanks!

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You may consult G.H Hardy's famous 'Inequalities'. Though pitched at a much higher level, it's an excellent text. You may obtain the pdf from here:- https://archive.org/details/in.ernet.dli.2015.278938