On two theorems of Hardy and Littlewood concerning moment constants (Q1344413)

Let \(\phi\) be a nonnegative real function defined on \([0,+\infty)\), \(\phi(0)= 0\), \(\phi(t)/t^ p\) is nonincreasing and \(\phi(t)/t^ q\) is nondecreasing with \(p\geq q>0\). Further, let \(h(s)\) be a positive nondecreasing function defined for \(1\leq s< \infty\) with the properties: \[ 1< \liminf_{s\to \infty} {h(2s)\over h(s)}\leq \limsup_{s\to \infty} {h(2s)\over h(s)}< K(h)< \infty. \] Then there exists a positive constant \(K= K(h,\phi)\) such that \[ \begin{multlined} K^{-1} \sum^ \infty_{m= 0} h(2^ m) \phi\left(\int_{I_ m} a(t) dt\right)\leq\\ \phi(A_ 0)+ \sum^ \infty_{n= 1} h(n) n^{-1}\phi(A_ n)\leq K \sum^ \infty_{m=0} h(2^ m) \phi\left(\int_{I_ m} a(t)dt\right),\end{multlined} \] where \(a_ n\geq 0\), \(a(t)\geq 0\), \(q_ n:= 1- 2^{-n}\), \(I_ n:= [q_ n, q_{n+1}]\), \(A(t):= \sum^ \infty_{n=1} a_ n t^ n\), \(t\in [0, 1]\), \(A_ n:= \int^ 1_ 0 a(t) t^ n dt\).