On Opial’s inequality for $f^ {(n)}$

Abstract:

We prove inequalities of the type \[ {\int _0^h {|{f^{(i)}}(x){f^{(j)}}(x)|dx \leq C(n,i,j,p){h^{2n - i - j + 1 - 2/p}}\left ( {\int _0^h {|{f^{(n)}}(x){|^p}dx} } \right )} ^{2/p}}\] when $f(0) = f’(0) = \cdots = {f^{(n - 1)}}(0) = 0$. We assume that ${f^{(n - 1)}}$ is absolutely continuous and ${f^{(n)}} \in {L_p}(0,h)$, with $p \geq 1,n \geq 2$, and $0 \leq i \leq j \leq n - 1$.