Summary: If \(p\) is a positive number, let \(A(p)\) denote a class of functions \(f(z) = z^{p} + \sum\limits_{k=p+1}^{\infty} a_{k} z^{k}\), which are analytic functions in the unit disc \(E\). Give a certain complex number \(\lambda \neq -p\) and \(f(z) \in A(p)\), we define operators \(J_{\lambda}\) by \(J_{\lambda} f(z) = h_{\lambda} (z) \ast f(z)\), where \(h_{\lambda} (z) = \sum\limits_{k=p}^{\infty} \frac{p+\lambda}{k+\lambda} z^{k}\).
In the paper, we obtain a conclusion that when \(J_{\lambda} f(z) \in R_{n}^{(p)} (\alpha) \, (0 \leq \alpha < p)\) there must be a number \(r_0\) so that as \(|z| < r_{0},\, f(z) \in R_{n}^{(p)} (\beta),\, 0 \leq \beta < p\).