  Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of normalized
analytic functions $f$ defined in the domain $|z|<1$ satisfying \begin{align*}
{\rm Re\,}
e^{i\phi}\left(\dfrac{}{}(1\!-\!\alpha\!+\!2\gamma)\!\left({f}/{z}\right)^\delta
+\left(\alpha\!-\!3\gamma+\gamma\left[\dfrac{}{}\left(1-{1}/{\delta}\right)\left({zf'}/{f}\right)+
{1}/{\delta}\left(1+{zf''}/{f'}\right)\right]\right)\right.\\
\left.\dfrac{}{}\left({f}/{z}\right)^\delta
\!\left({zf'}/{f}\right)-\beta\right)>0, \end{align*} with the conditions
$\alpha\geq 0$, $\beta<1$, $\gamma\geq 0$, $\delta>0$ and $\phi\in\mathbb{R}$.
Moreover, for $0<\delta\leq\frac{1}{(1-\zeta)}$, $0\leq\zeta<1$, the class
$\mathcal{C}_\delta(\zeta)$ be the subclass of normalized analytic functions
such that \begin{align*} {\rm
Re}{\,}\left(1/\delta\left(1+zf''/f'\right)+(1-1/\delta)\left({zf'}/{f}\right)\right)>\zeta,\quad
|z|<1. \end{align*} In the present work, the sufficient conditions on
$\lambda(t)$ are investigated, so that the generalized integral transform
\begin{align*} V_{\lambda}^\delta(f)(z)= \left(\int_0^1 \lambda(t)
\left({f(tz)}/{t}\right)^\delta dt\right)^{1/\delta},\quad |z|<1, \end{align*}
carries the functions from $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ into
$\mathcal{C}_\delta(\zeta)$. Several interesting applications are provided for
special choices of $\lambda(t)$.
