Sharp logarithmic inequalities for Hardy operators (Q2634627)

Summary: Let \(\ell\geq 1\) be a fixed number. We determine, for each \(K>0\), the best constant \(L=L(K,\ell)\in (0,\infty]\) such that the following holds: if \(f\) is a function on \((0,1]\) with \(\int_0^1 |f(r)|\,\text{d}r=1\), then \[ \int_0^1 t^{\ell-1}\left(\frac{1}{t}\int_0^t |f(r)|\,\text{d}r\right)^\ell\,\text{d}t\leq K \int_0^1 |f(r)|\log |f(r)|\,\text{d}r+L. \] As an application, we derive a sharp local logarithmic estimate for the \(n\)-dimensional fractional Hardy operator.