A note on certain inequalities related to Hardy's inequality
Using some ideas of \textit{N. Levinson} [Duke Math. J. 31, 389-394 (1964; Zbl 0126.281)], the author establishes conditions which are sufficient for the validity of the Hardy inequality \[ \int^ b_ aF^ p(x){dx\over x}\leq C\int^ b_ a[(\log x)f(x)]^ p{dx\over x}, \tag{1} \] where \(F(x)={1\over r(x)}\int_{I_ x}r(t)f(t){dt\over t}\) and \((a,b)=(1,b)\), \(I_ x=(x,b)\) with \(1<b\leq\infty\) or \((a,b)=(0,1)\), \(I_ x=(0,x)\). Let us note that, using the well-known Muckenhoupt condition [cf. \textit{B. Muckenhoupt} [Stud. Math. 44, 31-38 (1972; Zbl 0236.26015)], one can very easily find conditions which are both necessary and sufficient for the validity of (1) [cf. Section 11.1 of \textit{B. Opic} and \textit{A. Kufner}: ``Hardy-type inequalities'' (1990; Zbl 0698.26007)].
scientific article; zbMATH DE number 146995