On some new inequalities related to Hardy's integral inequality (Q919120)

A typical result is the following: \[ (\int^{\infty}_{0}z'(x)z(x)^{-m}w(x)F_ N(x)^ pdx)^{1/p}\leq p^ N| m-1|^{-N}\prod^{\infty}_{n=1}\alpha_ n(\int^{\infty}_{0}z'(x)z(x)^{-m}w(x)F_ 0(x)^ pdx)^{1/p}, \] where \(F_ 0(x)=z(x)f(x)\), \(F_ n(x)=z(x)I_ n\circ I_{n- 1}\circ...\circ I_ 1f(x)\) \((n=1,...,N)\), f measurable and nonnegative on \({\mathbb{R}}_+\), z differentiable there with positive derivative and with \(z(0+)>0\), w and \(r_ k\) are positive and locally absolutely continuous on \({\mathbb{R}}_+\), and \[ I_ kf(x)=\int^{x}_{0}r_ k(t)z'(t)f(t)dt/(r_ k(x)z(x))\quad (k=1,2,...,N), \] \[ \lim_{x\to 0+}(w(x)z(x)^{1-m}F_ n(x)^ p)=0,\quad m>1,\quad p\geq 1, \] \[ 0<1/\alpha_ n\leq \inf_{x\in {\mathbb{R}}_+}(1+(m-1)^{-1}z(x)z'(x)^{- 1}(pr_ n'(x)r_ n(x)^{-1}-w'(x)w(x)^{-1})). \]