The maximal function along a curve $(t,\gamma(t),t\gamma(t))$ on the Heisenberg group is discussed. The $L\sp p$-boundedness of this operator is shown under the doubling condition of $\gamma\sp \prime$ for convex $\gamma$ in $\mathbb {R}\sp +$. This condition also applies to the singular integrals when $\gamma$ is extended as an even or odd function. The proof is based on angular Littlewood-Paley decompositions in the Heisenberg group.