A Hardy-Littlewood integral inequality on finite intervals with a concave weight
The authors derive and prove that for all concave functions \( w:[a,b]\rightarrow [ 0,\infty)\) and for all functions \(f\in C^{2}[a,b] \) with \(f(a)=f(b)=0\) that the following inequality holds \[ \left( \int\nolimits_{a}^{b}w(x)f^{\prime}(x)^{2}dx\right) ^{2}\leq \left( \int\nolimits_{a}^{b}w(x)f(x)^{2}dx\right) \left( \int\nolimits_{a}^{b}w(x)f^{^{\prime \prime}}(x)^{2}dx\right) .\tag{1} \] Furthermore, an example is constructed with the help of MAPLE software to show that the concavity assumption above cannot be dropped and that (1) does not hold for convex weight functions.
scientific article; zbMATH DE number 6656709
scientific article