Let $\mathcal{A}$ be the class of normalized analytic functions in the unit disc
and let $P_{\gamma}(\alpha, \beta)$ be the class of all functions $f \in
\mathcal{A}$ satisfying the condition
\[ \exists \ \eta \in \mathbb{R}, \quad \Re \left \{ e^{i
\eta}\left[(1-\gamma)\left(\frac{f(z)}{z}\right)^{\alpha} + \gamma
\frac{zf'(z)}{f(z)}\left(\frac{f(z)}{z}\right)^{\alpha} - \beta
\right] \right \}
0 .\] We consider the integral transform \[ V_{\lambda,
\alpha}(f)(z)=\left\{\int_{0}^{1}\lambda(t) \left(\frac{f(tz)}{t}
\right)^{\alpha} dt\right\}^{\frac{1}{\alpha}},\] where $\lambda(t)$
is a real-valued nonnegative weight function normalized by\linebreak
$\int_{0}^{1}\lambda(t) dt=1$. In this paper we find conditions on
the parameters $\alpha, \beta, \gamma, \mu $ such that $V_{\lambda,
\alpha}(f)$ maps $P_{\gamma}(\alpha, \beta)$ into the class of
starlike functions of order $\mu$. We also provide a number of
applications for various choices of $\lambda(t)$. Our results
generalize known results on this topic.