On a theorem of Hardy and Littlewood

Abstract:

In this paper, we give an extension of a classical theorem of Hardy and Littlewood on power series. Let $\varphi$ be a strictly positive function defined on some interval $\left ( {\delta ,1} \right )$, satisfying a certain condition of limit. We prove that if $f\left ( x \right )$ is the sum of a convergent power series for $0 < x < 1$ with nonnegative coefficients ${a_n}$ and $f\left ( x \right ) \sim \varphi \left ( x \right )\;\left ( {x \to 1} \right )$, then ${S_n} \sim \alpha \cdot \varphi \left ( {x_0^{1/n}} \right )\left ( {n \to \infty } \right )$, where ${S_n} = {a_0} + {a_1} + \cdots + {a_{n,\;}}{x_0} \in \left ( {0,1} \right )$ and $\alpha$ depends only upon $\varphi$.