Suppose that \(f\) is a real continuous function and let \(s(x)=\int_{0}^{x}f(t)\,dt\). The Cesàro mean of \(s(x)\) is given by \(\sigma (s)(x)=x^{-1}\int_{0}^{x}s(t)\,dt\). If \(s(x)\rightarrow L\), then automatically \(\sigma (s)(x)\rightarrow L\), i.e., \(s(x)\) is Cesàro summable to \(L\). It is well known that the converse statement is false in general. A classical theorem of Hardy and Littlewood states that \(\sigma (s)(x)\rightarrow L\) together with the Tauberian condition \(xf(x)=O(1)\) imply that \( s(x)\rightarrow L\). In this paper, the authors prove several generalizations of this result.