Summary: Let \(H(\text{U})\) be the space of all analytic functions in the unit disk U and let \(\mathcal K\subset H(\text{U})\). For the operator \(A_{\beta,\gamma}:\mathcal K\to H(\text{U})\) defined by \[ A_{\beta,\gamma}(f)(z)=\left[\frac{\beta+\gamma}{z^\gamma} \int^z_0f^\beta (t)t^{\gamma-1}dt\right]^{1/\beta} \] and \(\beta,\gamma\in \mathbb C\), we will improve the result of T. Bulboaca [Bull. Korean Math. Soc., 34, 627-636 (1997; Zbl 0898.30021)] by determining weaker conditions on \(g\), \(\beta\) and \(\gamma\) such that \[ z\left[\frac{f(z)}z\right]^\beta \prec z\left[\frac{g(z)}z\right]^\beta\quad \text{implies}\quad z\left[\frac{A_{\beta,\gamma}(f)(z)}z\right]^\beta\prec z\left[ \frac{A_{\beta,\gamma}(g)(z)}z\right]^\beta, \] and this result is sharp, in the sense that the right-hand side function is the best dominant. In addition, we will give some particular cases of our main result obtained for appropriate choices of the parameters.