A Hardy-Littlewood-type theorem (Q1110027)

Let \(p(x)>0\) be a function integrable over [0,b] for any \(b>0\) such that \(x^{1-\alpha}p(x)\) is increasing for some \(\alpha >0\) and \(p(tx)/p(x)\to L(t)\) as \(x\to \infty\), for \(t\in [1/2,2]\), where L(t) is continuous at \(t=1\). Denote the class of all such functions p by HL. For p in HL, \(p^*(x)=\int^{\infty}_{0}p(u)e^{-u/x}du\) exists for \(x>0\). Assume that the function S(x) has the property that \(S^*(x)=\int^{\infty}_{0}S(u)e^{-u/x}du\) exists for \(x>0\). The author gives several sufficient conditions for the relation \(\lim_{x\to \infty}[S^*(x)/p^*(x)-S(x)/p(x)]=0\) to hold or the \(\lim it\lim_{x\to \infty}f(x)\) to exist, where \(f(x)=S^*(x)/p^*(x).\)