In this paper, using a generalized translation operator, we obtain an analog of Younis Theorem 5.2 in [M. S. Younis, Fourier transforms of Dini-Lipschitz functions, Int. J. Math. Math. Sci. 9 (2),(1986), 301--312.] for the Helgason Fourier transform of a set of functions satisfying the $(\eta,\gamma,k,2)$-Dini-Lipschitz condition in the space $L^{2}$ for functions on noncompact rank one Riemannian symmetric spaces.