On new generalizations of Hardy's integral inequality
The well-known Hardy's inequality states that \[ \int^\infty_0 \Biggl[x^{-1} \int^x_0 f(t)dt\Biggr]^p dx< q^p \int^\infty_0 f^p(t)dt\tag{1} \] provided that \(p\in (1,\infty)\), \(q= p/(p- 1)\), \(f>0\) and \(f\not\equiv 0\). The inequality (1) implies that for any \(a\) and \(b\) satisfying \(0<a<b<\infty\), \[ \int^b_a \Biggl[x^{-1} \int^x_a f(t)dt\Biggr]^p< q^p \int^b_a f^p(t)dt,\quad f>0,\quad f\not\equiv 0.\tag{2} \] One of the main results of the paper shows that the constant \(q^p\) in (2) can be replaced by \(q^p[1-(a/b)^{1/q}]^p\). The rest of the paper represents other variants of (1).
scientific article; zbMATH DE number 1115472