Convexity of the Generalized Integral Transform and Duality Techniques

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 $\lambda(t)$.